id	sid	tid	token	lemma	pos
ejpam-5587	1	1	european	european	PROPN
ejpam-5587	1	2	journal	journal	PROPN
ejpam-5587	1	3	of	of	ADP
ejpam-5587	1	4	pure	pure	ADJ
ejpam-5587	1	5	and	and	CCONJ
ejpam-5587	1	6	applied	applied	ADJ
ejpam-5587	1	7	mathematics	mathematic	NOUN
ejpam-5587	1	8	2025	2025	NUM
ejpam-5587	1	9	,	,	PUNCT
ejpam-5587	1	10	vol	vol	NOUN
ejpam-5587	1	11	.	.	PROPN
ejpam-5587	1	12	18	18	NUM
ejpam-5587	1	13	,	,	PUNCT
ejpam-5587	1	14	issue	issue	NOUN
ejpam-5587	1	15	1	1	NUM
ejpam-5587	1	16	,	,	PUNCT
ejpam-5587	1	17	article	article	NOUN
ejpam-5587	1	18	number	number	NOUN
ejpam-5587	1	19	5587	5587	NUM
ejpam-5587	1	20	issn	issn	PROPN
ejpam-5587	1	21	1307	1307	NUM
ejpam-5587	1	22	-	-	SYM
ejpam-5587	1	23	5543	5543	NUM
ejpam-5587	1	24	–	–	PUNCT
ejpam-5587	1	25	ejpam.com	ejpam.com	X
ejpam-5587	1	26	published	publish	VERB
ejpam-5587	1	27	by	by	ADP
ejpam-5587	1	28	new	new	PROPN
ejpam-5587	1	29	york	york	PROPN
ejpam-5587	1	30	business	business	PROPN
ejpam-5587	1	31	global	global	PROPN
ejpam-5587	1	32	on	on	ADP
ejpam-5587	1	33	the	the	DET
ejpam-5587	1	34	closability	closability	NOUN
ejpam-5587	1	35	of	of	ADP
ejpam-5587	1	36	class	class	NOUN
ejpam-5587	1	37	totally	totally	ADV
ejpam-5587	1	38	paranormal	paranormal	ADJ
ejpam-5587	1	39	operators	operator	NOUN
ejpam-5587	1	40	salam	salam	PROPN
ejpam-5587	1	41	alnabulsi1	alnabulsi1	PROPN
ejpam-5587	1	42	,	,	PUNCT
ejpam-5587	1	43	m.h.m	m.h.m	PROPN
ejpam-5587	1	44	.	.	PROPN
ejpam-5587	1	45	rashid2,∗	rashid2,∗	PROPN
ejpam-5587	1	46	1	1	NUM
ejpam-5587	1	47	department	department	NOUN
ejpam-5587	1	48	of	of	ADP
ejpam-5587	1	49	mathematics	mathematic	NOUN
ejpam-5587	1	50	,	,	PUNCT
ejpam-5587	1	51	faculty	faculty	NOUN
ejpam-5587	1	52	of	of	ADP
ejpam-5587	1	53	science	science	NOUN
ejpam-5587	1	54	,	,	PUNCT
ejpam-5587	1	55	university	university	PROPN
ejpam-5587	1	56	of	of	ADP
ejpam-5587	1	57	jordan	jordan	PROPN
ejpam-5587	1	58	,	,	PUNCT
ejpam-5587	1	59	amman	amman	PROPN
ejpam-5587	1	60	11942	11942	NUM
ejpam-5587	1	61	,	,	PUNCT
ejpam-5587	1	62	jordan	jordan	PROPN
ejpam-5587	1	63	2	2	NUM
ejpam-5587	1	64	department	department	NOUN
ejpam-5587	1	65	of	of	ADP
ejpam-5587	1	66	mathematics	mathematic	NOUN
ejpam-5587	1	67	,	,	PUNCT
ejpam-5587	1	68	faculty	faculty	NOUN
ejpam-5587	1	69	of	of	ADP
ejpam-5587	1	70	science	science	NOUN
ejpam-5587	1	71	p.o.box	p.o.box	PROPN
ejpam-5587	1	72	7	7	NUM
ejpam-5587	1	73	,	,	PUNCT
ejpam-5587	1	74	mutah	mutah	PROPN
ejpam-5587	1	75	university	university	PROPN
ejpam-5587	1	76	,	,	PUNCT
ejpam-5587	1	77	al	al	PROPN
ejpam-5587	1	78	-	-	PUNCT
ejpam-5587	1	79	karak	karak	PROPN
ejpam-5587	1	80	,	,	PUNCT
ejpam-5587	1	81	jordan	jordan	PROPN
ejpam-5587	1	82	abstract	abstract	PROPN
ejpam-5587	1	83	.	.	PUNCT
ejpam-5587	2	1	this	this	DET
ejpam-5587	2	2	article	article	NOUN
ejpam-5587	2	3	explores	explore	VERB
ejpam-5587	2	4	the	the	DET
ejpam-5587	2	5	analysis	analysis	NOUN
ejpam-5587	2	6	of	of	ADP
ejpam-5587	2	7	various	various	ADJ
ejpam-5587	2	8	spectral	spectral	ADJ
ejpam-5587	2	9	properties	property	NOUN
ejpam-5587	2	10	pertaining	pertain	VERB
ejpam-5587	2	11	to	to	ADP
ejpam-5587	2	12	totally	totally	ADV
ejpam-5587	2	13	paranormal	paranormal	VERB
ejpam-5587	2	14	closed	closed	ADJ
ejpam-5587	2	15	operators	operator	NOUN
ejpam-5587	2	16	,	,	PUNCT
ejpam-5587	2	17	extending	extend	VERB
ejpam-5587	2	18	beyond	beyond	ADP
ejpam-5587	2	19	the	the	DET
ejpam-5587	2	20	confines	confine	NOUN
ejpam-5587	2	21	of	of	ADP
ejpam-5587	2	22	boundedness	boundedness	NOUN
ejpam-5587	2	23	and	and	CCONJ
ejpam-5587	2	24	encompassing	encompass	VERB
ejpam-5587	2	25	operators	operator	NOUN
ejpam-5587	2	26	defined	define	VERB
ejpam-5587	2	27	in	in	ADP
ejpam-5587	2	28	a	a	DET
ejpam-5587	2	29	hilbert	hilbert	NOUN
ejpam-5587	2	30	space	space	NOUN
ejpam-5587	2	31	.	.	PUNCT
ejpam-5587	3	1	within	within	ADP
ejpam-5587	3	2	this	this	DET
ejpam-5587	3	3	class	class	NOUN
ejpam-5587	3	4	,	,	PUNCT
ejpam-5587	3	5	closed	close	VERB
ejpam-5587	3	6	symmetric	symmetric	ADJ
ejpam-5587	3	7	operators	operator	NOUN
ejpam-5587	3	8	are	be	AUX
ejpam-5587	3	9	included	include	VERB
ejpam-5587	3	10	.	.	PUNCT
ejpam-5587	4	1	initially	initially	ADV
ejpam-5587	4	2	,	,	PUNCT
ejpam-5587	4	3	we	we	PRON
ejpam-5587	4	4	establish	establish	VERB
ejpam-5587	4	5	that	that	SCONJ
ejpam-5587	4	6	the	the	DET
ejpam-5587	4	7	spectrum	spectrum	NOUN
ejpam-5587	4	8	of	of	ADP
ejpam-5587	4	9	such	such	DET
ejpam-5587	4	10	an	an	DET
ejpam-5587	4	11	operator	operator	NOUN
ejpam-5587	4	12	is	be	AUX
ejpam-5587	4	13	non	non	ADJ
ejpam-5587	4	14	-	-	ADJ
ejpam-5587	4	15	empty	empty	ADJ
ejpam-5587	4	16	and	and	CCONJ
ejpam-5587	4	17	provide	provide	VERB
ejpam-5587	4	18	a	a	DET
ejpam-5587	4	19	characterization	characterization	NOUN
ejpam-5587	4	20	of	of	ADP
ejpam-5587	4	21	closed	closed	ADJ
ejpam-5587	4	22	-	-	PUNCT
ejpam-5587	4	23	range	range	NOUN
ejpam-5587	4	24	operators	operator	NOUN
ejpam-5587	4	25	in	in	ADP
ejpam-5587	4	26	terms	term	NOUN
ejpam-5587	4	27	of	of	ADP
ejpam-5587	4	28	the	the	DET
ejpam-5587	4	29	spectrum	spectrum	NOUN
ejpam-5587	4	30	.	.	PUNCT
ejpam-5587	5	1	building	build	VERB
ejpam-5587	5	2	on	on	ADP
ejpam-5587	5	3	these	these	DET
ejpam-5587	5	4	findings	finding	NOUN
ejpam-5587	5	5	,	,	PUNCT
ejpam-5587	5	6	we	we	PRON
ejpam-5587	5	7	proceed	proceed	VERB
ejpam-5587	5	8	to	to	PART
ejpam-5587	5	9	prove	prove	VERB
ejpam-5587	5	10	weyl	weyl	PROPN
ejpam-5587	5	11	’s	’s	PART
ejpam-5587	5	12	theorem	theorem	NOUN
ejpam-5587	5	13	,	,	PUNCT
ejpam-5587	5	14	demonstrating	demonstrate	VERB
ejpam-5587	5	15	that	that	PRON
ejpam-5587	5	16	for	for	ADP
ejpam-5587	5	17	a	a	DET
ejpam-5587	5	18	densely	densely	ADV
ejpam-5587	5	19	defined	define	VERB
ejpam-5587	5	20	closed	close	VERB
ejpam-5587	5	21	totally	totally	ADV
ejpam-5587	5	22	paranormal	paranormal	ADJ
ejpam-5587	5	23	operator	operator	NOUN
ejpam-5587	5	24	t	t	NOUN
ejpam-5587	5	25	,	,	PUNCT
ejpam-5587	5	26	the	the	DET
ejpam-5587	5	27	difference	difference	NOUN
ejpam-5587	5	28	between	between	ADP
ejpam-5587	5	29	the	the	DET
ejpam-5587	5	30	spectrum	spectrum	NOUN
ejpam-5587	5	31	σ(t	σ(t	PROPN
ejpam-5587	5	32	)	)	PUNCT
ejpam-5587	5	33	and	and	CCONJ
ejpam-5587	5	34	the	the	DET
ejpam-5587	5	35	weyl	weyl	PROPN
ejpam-5587	5	36	spectrum	spectrum	NOUN
ejpam-5587	5	37	σw(t	σw(t	PUNCT
ejpam-5587	5	38	)	)	PUNCT
ejpam-5587	5	39	equals	equal	VERB
ejpam-5587	5	40	the	the	DET
ejpam-5587	5	41	set	set	NOUN
ejpam-5587	5	42	of	of	ADP
ejpam-5587	5	43	all	all	DET
ejpam-5587	5	44	isolated	isolate	VERB
ejpam-5587	5	45	eigenvalues	eigenvalue	NOUN
ejpam-5587	5	46	with	with	ADP
ejpam-5587	5	47	finite	finite	PROPN
ejpam-5587	5	48	multiplicities	multiplicity	NOUN
ejpam-5587	5	49	,	,	PUNCT
ejpam-5587	5	50	denoted	denote	VERB
ejpam-5587	5	51	by	by	ADP
ejpam-5587	5	52	π00(t	π00(t	PROPN
ejpam-5587	5	53	)	)	PUNCT
ejpam-5587	5	54	.	.	PUNCT
ejpam-5587	6	1	in	in	ADP
ejpam-5587	6	2	the	the	DET
ejpam-5587	6	3	final	final	ADJ
ejpam-5587	6	4	section	section	NOUN
ejpam-5587	6	5	,	,	PUNCT
ejpam-5587	6	6	we	we	PRON
ejpam-5587	6	7	establish	establish	VERB
ejpam-5587	6	8	the	the	DET
ejpam-5587	6	9	self	self	NOUN
ejpam-5587	6	10	-	-	PUNCT
ejpam-5587	6	11	adjointness	adjointness	NOUN
ejpam-5587	6	12	of	of	ADP
ejpam-5587	6	13	the	the	DET
ejpam-5587	6	14	riesz	riesz	PROPN
ejpam-5587	6	15	projection	projection	NOUN
ejpam-5587	6	16	eµ	eµ	NOUN
ejpam-5587	6	17	corresponding	correspond	VERB
ejpam-5587	6	18	to	to	ADP
ejpam-5587	6	19	any	any	DET
ejpam-5587	6	20	non	non	ADJ
ejpam-5587	6	21	-	-	ADJ
ejpam-5587	6	22	zero	zero	NUM
ejpam-5587	6	23	isolated	isolate	VERB
ejpam-5587	6	24	spectral	spectral	ADJ
ejpam-5587	6	25	value	value	NOUN
ejpam-5587	6	26	µ	µ	PROPN
ejpam-5587	6	27	of	of	ADP
ejpam-5587	6	28	t	t	PROPN
ejpam-5587	6	29	.	.	PUNCT
ejpam-5587	7	1	furthermore	furthermore	ADV
ejpam-5587	7	2	,	,	PUNCT
ejpam-5587	7	3	we	we	PRON
ejpam-5587	7	4	show	show	VERB
ejpam-5587	7	5	that	that	SCONJ
ejpam-5587	7	6	this	this	DET
ejpam-5587	7	7	riesz	riesz	NOUN
ejpam-5587	7	8	projection	projection	NOUN
ejpam-5587	7	9	satisfies	satisfy	VERB
ejpam-5587	7	10	the	the	DET
ejpam-5587	7	11	relationships	relationship	NOUN
ejpam-5587	7	12	ran(eµ	ran(eµ	NOUN
ejpam-5587	7	13	)	)	PUNCT
ejpam-5587	7	14	=	=	SYM
ejpam-5587	7	15	ker(t	ker(t	NOUN
ejpam-5587	7	16	−	−	NUM
ejpam-5587	7	17	µi	µi	NOUN
ejpam-5587	7	18	)	)	PUNCT
ejpam-5587	8	1	=	=	PUNCT
ejpam-5587	8	2	ker(t	ker(t	NOUN
ejpam-5587	8	3	−	−	NOUN
ejpam-5587	8	4	µi)∗.	µi)∗.	ADV
ejpam-5587	8	5	additionally	additionally	ADV
ejpam-5587	8	6	,	,	PUNCT
ejpam-5587	8	7	we	we	PRON
ejpam-5587	8	8	demonstrate	demonstrate	VERB
ejpam-5587	8	9	that	that	SCONJ
ejpam-5587	8	10	if	if	SCONJ
ejpam-5587	8	11	t	t	PROPN
ejpam-5587	8	12	is	be	AUX
ejpam-5587	8	13	a	a	DET
ejpam-5587	8	14	closed	closed	ADJ
ejpam-5587	8	15	totally	totally	ADV
ejpam-5587	8	16	paranormal	paranormal	ADJ
ejpam-5587	8	17	operator	operator	NOUN
ejpam-5587	8	18	with	with	ADP
ejpam-5587	8	19	a	a	DET
ejpam-5587	8	20	weyl	weyl	VERB
ejpam-5587	8	21	spectrum	spectrum	NOUN
ejpam-5587	8	22	σw(t	σw(t	PUNCT
ejpam-5587	8	23	)	)	PUNCT
ejpam-5587	8	24	=	=	SYM
ejpam-5587	8	25	0	0	NUM
ejpam-5587	8	26	,	,	PUNCT
ejpam-5587	8	27	then	then	ADV
ejpam-5587	8	28	t	t	PROPN
ejpam-5587	8	29	qualifies	qualify	VERB
ejpam-5587	8	30	as	as	ADP
ejpam-5587	8	31	a	a	DET
ejpam-5587	8	32	compact	compact	ADJ
ejpam-5587	8	33	normal	normal	ADJ
ejpam-5587	8	34	operator	operator	NOUN
ejpam-5587	8	35	.	.	PUNCT
ejpam-5587	9	1	2020	2020	NUM
ejpam-5587	9	2	mathematics	mathematics	PROPN
ejpam-5587	9	3	subject	subject	NOUN
ejpam-5587	9	4	classifications	classification	NOUN
ejpam-5587	9	5	:	:	PUNCT
ejpam-5587	9	6	47a10,47a53	47a10,47a53	NUM
ejpam-5587	9	7	,	,	PUNCT
ejpam-5587	9	8	47b20	47b20	NUM
ejpam-5587	9	9	key	key	ADJ
ejpam-5587	9	10	words	word	NOUN
ejpam-5587	9	11	and	and	CCONJ
ejpam-5587	9	12	phrases	phrase	NOUN
ejpam-5587	9	13	:	:	PUNCT
ejpam-5587	9	14	densely	densely	ADV
ejpam-5587	9	15	defined	define	VERB
ejpam-5587	9	16	operator	operator	NOUN
ejpam-5587	9	17	,	,	PUNCT
ejpam-5587	9	18	closed	closed	ADJ
ejpam-5587	9	19	operator	operator	NOUN
ejpam-5587	9	20	,	,	PUNCT
ejpam-5587	9	21	totally	totally	ADV
ejpam-5587	9	22	paranormal	paranormal	ADJ
ejpam-5587	9	23	,	,	PUNCT
ejpam-5587	9	24	reduced	reduce	VERB
ejpam-5587	9	25	minimum	minimum	ADJ
ejpam-5587	9	26	modulus	modulus	NOUN
ejpam-5587	9	27	,	,	PUNCT
ejpam-5587	9	28	riesz	riesz	NOUN
ejpam-5587	9	29	projection	projection	NOUN
ejpam-5587	9	30	,	,	PUNCT
ejpam-5587	9	31	weyl	weyl	PROPN
ejpam-5587	9	32	’s	’s	PART
ejpam-5587	9	33	theorem	theorem	ADJ
ejpam-5587	9	34	1	1	NUM
ejpam-5587	9	35	.	.	PUNCT
ejpam-5587	10	1	introduction	introduction	NOUN
ejpam-5587	10	2	the	the	DET
ejpam-5587	10	3	class	class	NOUN
ejpam-5587	10	4	of	of	ADP
ejpam-5587	10	5	normal	normal	ADJ
ejpam-5587	10	6	operators	operator	NOUN
ejpam-5587	10	7	is	be	AUX
ejpam-5587	10	8	fundamental	fundamental	ADJ
ejpam-5587	10	9	in	in	ADP
ejpam-5587	10	10	operator	operator	NOUN
ejpam-5587	10	11	theory	theory	NOUN
ejpam-5587	10	12	,	,	PUNCT
ejpam-5587	10	13	having	having	AUX
ejpam-5587	10	14	been	be	AUX
ejpam-5587	10	15	the	the	DET
ejpam-5587	10	16	subject	subject	NOUN
ejpam-5587	10	17	of	of	ADP
ejpam-5587	10	18	significant	significant	ADJ
ejpam-5587	10	19	research	research	NOUN
ejpam-5587	10	20	.	.	PUNCT
ejpam-5587	11	1	the	the	DET
ejpam-5587	11	2	spectral	spectral	ADJ
ejpam-5587	11	3	theorem	theorem	NOUN
ejpam-5587	11	4	for	for	ADP
ejpam-5587	11	5	these	these	DET
ejpam-5587	11	6	operators	operator	NOUN
ejpam-5587	11	7	confirms	confirm	VERB
ejpam-5587	11	8	the	the	DET
ejpam-5587	11	9	existence	existence	NOUN
ejpam-5587	11	10	of	of	ADP
ejpam-5587	11	11	non	non	ADJ
ejpam-5587	11	12	-	-	ADJ
ejpam-5587	11	13	trivial	trivial	ADJ
ejpam-5587	11	14	invariant	invariant	ADJ
ejpam-5587	11	15	subspaces	subspace	NOUN
ejpam-5587	11	16	and	and	CCONJ
ejpam-5587	11	17	provides	provide	VERB
ejpam-5587	11	18	insight	insight	NOUN
ejpam-5587	11	19	into	into	ADP
ejpam-5587	11	20	the	the	DET
ejpam-5587	11	21	operator	operator	NOUN
ejpam-5587	11	22	’s	’s	PART
ejpam-5587	11	23	full	full	ADJ
ejpam-5587	11	24	structure	structure	NOUN
ejpam-5587	11	25	.	.	PUNCT
ejpam-5587	12	1	the	the	DET
ejpam-5587	12	2	category	category	NOUN
ejpam-5587	12	3	of	of	ADP
ejpam-5587	12	4	bounded	bounded	ADJ
ejpam-5587	12	5	paranormal	paranormal	PROPN
ejpam-5587	12	6	operators	operator	NOUN
ejpam-5587	12	7	was	be	AUX
ejpam-5587	12	8	initially	initially	ADV
ejpam-5587	12	9	investigated	investigate	VERB
ejpam-5587	12	10	by	by	ADP
ejpam-5587	12	11	istrǎtescu	istrǎtescu	PROPN
ejpam-5587	12	12	,	,	PUNCT
ejpam-5587	12	13	who	who	PRON
ejpam-5587	12	14	referred	refer	VERB
ejpam-5587	12	15	to	to	ADP
ejpam-5587	12	16	it	it	PRON
ejpam-5587	12	17	as	as	ADP
ejpam-5587	12	18	class	class	NOUN
ejpam-5587	12	19	n	n	CCONJ
ejpam-5587	12	20	[	[	X
ejpam-5587	12	21	13	13	NUM
ejpam-5587	12	22	]	]	PUNCT
ejpam-5587	12	23	.	.	PUNCT
ejpam-5587	13	1	later	later	ADV
ejpam-5587	13	2	,	,	PUNCT
ejpam-5587	13	3	furuta	furuta	PROPN
ejpam-5587	13	4	coined	coin	VERB
ejpam-5587	13	5	the	the	DET
ejpam-5587	13	6	term	term	NOUN
ejpam-5587	13	7	”	"	PUNCT
ejpam-5587	13	8	paranormal	paranormal	ADJ
ejpam-5587	13	9	operator	operator	NOUN
ejpam-5587	13	10	”	"	PUNCT
ejpam-5587	14	1	[	[	X
ejpam-5587	14	2	8	8	NUM
ejpam-5587	14	3	]	]	PUNCT
ejpam-5587	14	4	.	.	PUNCT
ejpam-5587	15	1	numerous	numerous	ADJ
ejpam-5587	15	2	researchers	researcher	NOUN
ejpam-5587	15	3	have	have	AUX
ejpam-5587	15	4	since	since	SCONJ
ejpam-5587	15	5	studied	study	VERB
ejpam-5587	15	6	bounded	bounded	ADJ
ejpam-5587	15	7	paranormal	paranormal	PROPN
ejpam-5587	15	8	operators	operator	NOUN
ejpam-5587	15	9	,	,	PUNCT
ejpam-5587	15	10	including	include	VERB
ejpam-5587	15	11	∗corresponding	∗corresponde	VERB
ejpam-5587	15	12	author	author	NOUN
ejpam-5587	15	13	.	.	PUNCT
ejpam-5587	16	1	doi	doi	NOUN
ejpam-5587	16	2	:	:	PUNCT
ejpam-5587	16	3	https://doi.org/10.29020/nybg.ejpam.v18i1.5587	https://doi.org/10.29020/nybg.ejpam.v18i1.5587	PROPN
ejpam-5587	16	4	email	email	NOUN
ejpam-5587	16	5	addresses	address	VERB
ejpam-5587	16	6	:	:	PUNCT
ejpam-5587	16	7	s.alnabulsi@ju.edu.jo	s.alnabulsi@ju.edu.jo	PROPN
ejpam-5587	16	8	(	(	PUNCT
ejpam-5587	16	9	s.	s.	PROPN
ejpam-5587	16	10	alnabulsi	alnabulsi	PROPN
ejpam-5587	16	11	)	)	PUNCT
ejpam-5587	16	12	,	,	PUNCT
ejpam-5587	16	13	malik	malik	PROPN
ejpam-5587	16	14	okasha@yahoo.com	okasha@yahoo.com	X
ejpam-5587	16	15	(	(	PUNCT
ejpam-5587	16	16	m.h.m	m.h.m	PROPN
ejpam-5587	16	17	.	.	PROPN
ejpam-5587	16	18	rashid	rashid	PROPN
ejpam-5587	16	19	)	)	PUNCT
ejpam-5587	16	20	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5587	17	1	1	1	NUM
ejpam-5587	17	2	copyright	copyright	NOUN
ejpam-5587	17	3	:	:	PUNCT
ejpam-5587	17	4	©	©	PROPN
ejpam-5587	17	5	2025	2025	NUM
ejpam-5587	17	6	the	the	DET
ejpam-5587	17	7	author(s	author(s	NOUN
ejpam-5587	17	8	)	)	PUNCT
ejpam-5587	17	9	.	.	PUNCT
ejpam-5587	18	1	(	(	PUNCT
ejpam-5587	18	2	cc	cc	NOUN
ejpam-5587	18	3	by	by	ADP
ejpam-5587	18	4	-	-	PUNCT
ejpam-5587	18	5	nc	nc	PROPN
ejpam-5587	18	6	4.0	4.0	NUM
ejpam-5587	18	7	)	)	PUNCT
ejpam-5587	18	8	s.	s.	PROPN
ejpam-5587	18	9	alnabulsi	alnabulsi	PROPN
ejpam-5587	18	10	,	,	PUNCT
ejpam-5587	18	11	m.h.m	m.h.m	PROPN
ejpam-5587	18	12	.	.	PUNCT
ejpam-5587	18	13	rashid	rashid	PROPN
ejpam-5587	18	14	/	/	SYM
ejpam-5587	18	15	eur	eur	PROPN
ejpam-5587	18	16	.	.	PUNCT
ejpam-5587	19	1	j.	j.	PROPN
ejpam-5587	19	2	pure	pure	PROPN
ejpam-5587	19	3	appl	appl	PROPN
ejpam-5587	19	4	.	.	PROPN
ejpam-5587	19	5	math	math	PROPN
ejpam-5587	19	6	,	,	PUNCT
ejpam-5587	19	7	18	18	NUM
ejpam-5587	19	8	(	(	PUNCT
ejpam-5587	19	9	1	1	NUM
ejpam-5587	19	10	)	)	PUNCT
ejpam-5587	19	11	(	(	PUNCT
ejpam-5587	19	12	2025	2025	NUM
ejpam-5587	19	13	)	)	PUNCT
ejpam-5587	19	14	,	,	PUNCT
ejpam-5587	19	15	5587	5587	NUM
ejpam-5587	19	16	2	2	NUM
ejpam-5587	19	17	of	of	ADP
ejpam-5587	19	18	20	20	NUM
ejpam-5587	19	19	works	work	NOUN
ejpam-5587	19	20	by	by	ADP
ejpam-5587	19	21	ando	ando	PROPN
ejpam-5587	19	22	and	and	CCONJ
ejpam-5587	19	23	others	other	NOUN
ejpam-5587	20	1	[	[	X
ejpam-5587	20	2	3	3	NUM
ejpam-5587	20	3	,	,	PUNCT
ejpam-5587	20	4	8	8	NUM
ejpam-5587	20	5	,	,	PUNCT
ejpam-5587	20	6	13	13	NUM
ejpam-5587	20	7	]	]	PUNCT
ejpam-5587	20	8	.	.	PUNCT
ejpam-5587	21	1	specifically	specifically	ADV
ejpam-5587	21	2	,	,	PUNCT
ejpam-5587	21	3	ando	ando	PROPN
ejpam-5587	22	1	[	[	X
ejpam-5587	22	2	3	3	NUM
ejpam-5587	22	3	]	]	PUNCT
ejpam-5587	22	4	provided	provide	VERB
ejpam-5587	22	5	a	a	DET
ejpam-5587	22	6	characterization	characterization	NOUN
ejpam-5587	22	7	of	of	ADP
ejpam-5587	22	8	bounded	bounded	ADJ
ejpam-5587	22	9	paranormal	paranormal	PROPN
ejpam-5587	22	10	operators	operator	NOUN
ejpam-5587	22	11	,	,	PUNCT
ejpam-5587	22	12	while	while	SCONJ
ejpam-5587	22	13	istrǎtescu	istrǎtescu	NOUN
ejpam-5587	22	14	demonstrated	demonstrate	VERB
ejpam-5587	22	15	that	that	SCONJ
ejpam-5587	22	16	normaloid	normaloid	PROPN
ejpam-5587	22	17	operators	operator	NOUN
ejpam-5587	22	18	generalize	generalize	VERB
ejpam-5587	22	19	paranormal	paranormal	ADJ
ejpam-5587	22	20	operators	operator	NOUN
ejpam-5587	23	1	[	[	X
ejpam-5587	23	2	13	13	NUM
ejpam-5587	23	3	]	]	PUNCT
ejpam-5587	23	4	.	.	PUNCT
ejpam-5587	24	1	a	a	DET
ejpam-5587	24	2	continuous	continuous	ADJ
ejpam-5587	24	3	linear	linear	NOUN
ejpam-5587	24	4	operator	operator	NOUN
ejpam-5587	24	5	on	on	ADP
ejpam-5587	24	6	a	a	DET
ejpam-5587	24	7	complex	complex	ADJ
ejpam-5587	24	8	banach	banach	NOUN
ejpam-5587	24	9	space	space	NOUN
ejpam-5587	24	10	is	be	AUX
ejpam-5587	24	11	said	say	VERB
ejpam-5587	24	12	to	to	PART
ejpam-5587	24	13	be	be	AUX
ejpam-5587	24	14	paranormal	paranormal	ADJ
ejpam-5587	24	15	if	if	SCONJ
ejpam-5587	24	16	∥tx∥2	∥tx∥2	DET
ejpam-5587	24	17	≤	≤	NUM
ejpam-5587	24	18	∥∥t	∥∥t	VERB
ejpam-5587	24	19	2x	2x	NUM
ejpam-5587	24	20	∥∥	∥∥	PROPN
ejpam-5587	24	21	∥x∥	∥x∥	NOUN
ejpam-5587	24	22	for	for	ADP
ejpam-5587	24	23	all	all	DET
ejpam-5587	24	24	x	x	SYM
ejpam-5587	24	25	∈	∈	PROPN
ejpam-5587	24	26	x	x	NOUN
ejpam-5587	24	27	,	,	PUNCT
ejpam-5587	24	28	where	where	SCONJ
ejpam-5587	24	29	x	x	PRON
ejpam-5587	24	30	is	be	AUX
ejpam-5587	24	31	a	a	DET
ejpam-5587	24	32	banach	banach	NOUN
ejpam-5587	24	33	space	space	NOUN
ejpam-5587	24	34	.	.	PUNCT
ejpam-5587	25	1	t	t	PROPN
ejpam-5587	25	2	is	be	AUX
ejpam-5587	25	3	called	call	VERB
ejpam-5587	25	4	totally	totally	ADV
ejpam-5587	25	5	paranormal	paranormal	ADJ
ejpam-5587	25	6	[	[	X
ejpam-5587	25	7	22	22	NUM
ejpam-5587	25	8	]	]	PUNCT
ejpam-5587	25	9	if	if	SCONJ
ejpam-5587	25	10	t	t	PROPN
ejpam-5587	25	11	−	−	PROPN
ejpam-5587	25	12	µi	µi	PROPN
ejpam-5587	25	13	is	be	AUX
ejpam-5587	25	14	paranormal	paranormal	ADJ
ejpam-5587	25	15	for	for	ADP
ejpam-5587	25	16	every	every	DET
ejpam-5587	25	17	µ	µ	PROPN
ejpam-5587	25	18	∈	∈	PROPN
ejpam-5587	25	19	c.	c.	NOUN
ejpam-5587	25	20	that	that	PRON
ejpam-5587	25	21	is	be	AUX
ejpam-5587	25	22	,	,	PUNCT
ejpam-5587	25	23	∥(t	∥(t	VERB
ejpam-5587	25	24	−	−	ADP
ejpam-5587	25	25	µi)x∥2	µi)x∥2	NOUN
ejpam-5587	25	26	≤∥∥(t	≤∥∥(t	NOUN
ejpam-5587	25	27	−	−	PROPN
ejpam-5587	25	28	µi)2x	µi)2x	VERB
ejpam-5587	25	29	∥∥	∥∥	X
ejpam-5587	25	30	∥x∥	∥x∥	NOUN
ejpam-5587	25	31	for	for	ADP
ejpam-5587	25	32	all	all	DET
ejpam-5587	25	33	x	x	SYM
ejpam-5587	25	34	∈	∈	PROPN
ejpam-5587	25	35	x	x	X
ejpam-5587	25	36	and	and	CCONJ
ejpam-5587	25	37	µ	µ	PROPN
ejpam-5587	25	38	∈	∈	PROPN
ejpam-5587	25	39	c.	c.	NOUN
ejpam-5587	25	40	hence	hence	ADV
ejpam-5587	25	41	,	,	PUNCT
ejpam-5587	25	42	we	we	PRON
ejpam-5587	25	43	have	have	VERB
ejpam-5587	25	44	the	the	DET
ejpam-5587	25	45	following	follow	VERB
ejpam-5587	25	46	inclusion	inclusion	NOUN
ejpam-5587	25	47	relation	relation	NOUN
ejpam-5587	25	48	between	between	ADP
ejpam-5587	25	49	some	some	DET
ejpam-5587	25	50	subclasses	subclass	NOUN
ejpam-5587	25	51	and	and	CCONJ
ejpam-5587	25	52	a	a	DET
ejpam-5587	25	53	generalized	generalized	ADJ
ejpam-5587	25	54	class	class	NOUN
ejpam-5587	25	55	of	of	ADP
ejpam-5587	25	56	bounded	bound	VERB
ejpam-5587	25	57	totally	totally	ADV
ejpam-5587	25	58	paranormal	paranormal	ADJ
ejpam-5587	25	59	operators	operator	NOUN
ejpam-5587	25	60	.	.	PUNCT
ejpam-5587	26	1	normal	normal	ADJ
ejpam-5587	26	2	⊆	⊆	NUM
ejpam-5587	26	3	hyponormal	hyponormal	ADJ
ejpam-5587	26	4	⊆	⊆	NUM
ejpam-5587	26	5	totally	totally	ADV
ejpam-5587	26	6	paranormal	paranormal	ADJ
ejpam-5587	26	7	⊆	⊆	NUM
ejpam-5587	26	8	paranormal	paranormal	NOUN
ejpam-5587	26	9	⊆	⊆	NUM
ejpam-5587	26	10	normaloid	normaloid	NOUN
ejpam-5587	26	11	.	.	PUNCT
ejpam-5587	27	1	the	the	DET
ejpam-5587	27	2	inclusion	inclusion	NOUN
ejpam-5587	27	3	relationships	relationship	NOUN
ejpam-5587	27	4	mentioned	mention	VERB
ejpam-5587	27	5	above	above	ADV
ejpam-5587	27	6	are	be	AUX
ejpam-5587	27	7	strict	strict	ADJ
ejpam-5587	27	8	.	.	PUNCT
ejpam-5587	28	1	for	for	ADP
ejpam-5587	28	2	additional	additional	ADJ
ejpam-5587	28	3	information	information	NOUN
ejpam-5587	28	4	,	,	PUNCT
ejpam-5587	28	5	see	see	VERB
ejpam-5587	28	6	[	[	X
ejpam-5587	28	7	8	8	NUM
ejpam-5587	28	8	,	,	PUNCT
ejpam-5587	28	9	22	22	NUM
ejpam-5587	28	10	]	]	PUNCT
ejpam-5587	28	11	.	.	PUNCT
ejpam-5587	29	1	daniluk	daniluk	PROPN
ejpam-5587	29	2	extended	extend	VERB
ejpam-5587	29	3	the	the	DET
ejpam-5587	29	4	concept	concept	NOUN
ejpam-5587	29	5	of	of	ADP
ejpam-5587	29	6	bounded	bounded	ADJ
ejpam-5587	29	7	paranormal	paranormal	PROPN
ejpam-5587	29	8	operators	operator	NOUN
ejpam-5587	29	9	to	to	PART
ejpam-5587	29	10	encompass	encompass	VERB
ejpam-5587	29	11	unbounded	unbounded	ADJ
ejpam-5587	29	12	operators	operator	NOUN
ejpam-5587	29	13	,	,	PUNCT
ejpam-5587	29	14	exploring	explore	VERB
ejpam-5587	29	15	the	the	DET
ejpam-5587	29	16	conditions	condition	NOUN
ejpam-5587	29	17	for	for	ADP
ejpam-5587	29	18	their	their	PRON
ejpam-5587	29	19	closability	closability	NOUN
ejpam-5587	29	20	[	[	X
ejpam-5587	29	21	6	6	NUM
ejpam-5587	29	22	]	]	PUNCT
ejpam-5587	29	23	.	.	PUNCT
ejpam-5587	30	1	in	in	ADP
ejpam-5587	30	2	this	this	DET
ejpam-5587	30	3	paper	paper	NOUN
ejpam-5587	30	4	,	,	PUNCT
ejpam-5587	30	5	we	we	PRON
ejpam-5587	30	6	focus	focus	VERB
ejpam-5587	30	7	on	on	ADP
ejpam-5587	30	8	densely	densely	ADV
ejpam-5587	30	9	defined	define	VERB
ejpam-5587	30	10	,	,	PUNCT
ejpam-5587	30	11	closed	close	VERB
ejpam-5587	30	12	totally	totally	ADV
ejpam-5587	30	13	paranormal	paranormal	ADJ
ejpam-5587	30	14	operators	operator	NOUN
ejpam-5587	30	15	in	in	ADP
ejpam-5587	30	16	a	a	DET
ejpam-5587	30	17	hilbert	hilbert	NOUN
ejpam-5587	30	18	space	space	NOUN
ejpam-5587	30	19	h	h	NOUN
ejpam-5587	30	20	and	and	CCONJ
ejpam-5587	30	21	establish	establish	VERB
ejpam-5587	30	22	the	the	DET
ejpam-5587	30	23	following	follow	VERB
ejpam-5587	30	24	results	result	NOUN
ejpam-5587	30	25	.	.	PUNCT
ejpam-5587	31	1	let	let	VERB
ejpam-5587	31	2	t	t	NOUN
ejpam-5587	31	3	be	be	AUX
ejpam-5587	31	4	a	a	DET
ejpam-5587	31	5	densely	densely	ADV
ejpam-5587	31	6	defined	define	VERB
ejpam-5587	31	7	closed	close	VERB
ejpam-5587	31	8	totally	totally	ADV
ejpam-5587	31	9	paranormal	paranormal	ADJ
ejpam-5587	31	10	operator	operator	NOUN
ejpam-5587	31	11	in	in	ADP
ejpam-5587	31	12	h.	h.	PROPN
ejpam-5587	31	13	then	then	ADV
ejpam-5587	31	14	(	(	PUNCT
ejpam-5587	31	15	i	i	NOUN
ejpam-5587	31	16	)	)	PUNCT
ejpam-5587	31	17	spectrum	spectrum	NOUN
ejpam-5587	31	18	of	of	ADP
ejpam-5587	31	19	t	t	PROPN
ejpam-5587	31	20	is	be	AUX
ejpam-5587	31	21	non	non	ADJ
ejpam-5587	31	22	-	-	ADJ
ejpam-5587	31	23	empty	empty	ADJ
ejpam-5587	31	24	.	.	PUNCT
ejpam-5587	32	1	(	(	PUNCT
ejpam-5587	32	2	ii	ii	NOUN
ejpam-5587	32	3	)	)	PUNCT
ejpam-5587	32	4	every	every	DET
ejpam-5587	32	5	isolated	isolate	VERB
ejpam-5587	32	6	spectral	spectral	ADJ
ejpam-5587	32	7	value	value	NOUN
ejpam-5587	32	8	of	of	ADP
ejpam-5587	32	9	t	t	PROPN
ejpam-5587	32	10	is	be	AUX
ejpam-5587	32	11	an	an	DET
ejpam-5587	32	12	eigenvalue	eigenvalue	NOUN
ejpam-5587	32	13	.	.	PUNCT
ejpam-5587	33	1	(	(	PUNCT
ejpam-5587	33	2	iii	iii	NOUN
ejpam-5587	33	3	)	)	PUNCT
ejpam-5587	33	4	in	in	ADP
ejpam-5587	33	5	addition	addition	NOUN
ejpam-5587	33	6	,	,	PUNCT
ejpam-5587	33	7	if	if	SCONJ
ejpam-5587	33	8	ker(t	ker(t	NOUN
ejpam-5587	33	9	)	)	PUNCT
ejpam-5587	34	1	=	=	SYM
ejpam-5587	34	2	ker(t	ker(t	NOUN
ejpam-5587	34	3	∗	∗	NOUN
ejpam-5587	34	4	)	)	PUNCT
ejpam-5587	34	5	,	,	PUNCT
ejpam-5587	34	6	then	then	ADV
ejpam-5587	34	7	(	(	PUNCT
ejpam-5587	34	8	a	a	X
ejpam-5587	34	9	)	)	PUNCT
ejpam-5587	34	10	range	range	NOUN
ejpam-5587	34	11	of	of	ADP
ejpam-5587	34	12	t	t	PROPN
ejpam-5587	34	13	is	be	AUX
ejpam-5587	34	14	closed	close	VERB
ejpam-5587	34	15	if	if	SCONJ
ejpam-5587	34	16	and	and	CCONJ
ejpam-5587	34	17	only	only	ADV
ejpam-5587	34	18	if	if	SCONJ
ejpam-5587	34	19	0	0	NUM
ejpam-5587	34	20	is	be	AUX
ejpam-5587	34	21	an	an	DET
ejpam-5587	34	22	isolated	isolated	ADJ
ejpam-5587	34	23	spectral	spectral	ADJ
ejpam-5587	34	24	value	value	NOUN
ejpam-5587	34	25	of	of	ADP
ejpam-5587	34	26	t	t	PROPN
ejpam-5587	34	27	.	.	PUNCT
ejpam-5587	35	1	(	(	PUNCT
ejpam-5587	35	2	b	b	X
ejpam-5587	35	3	)	)	PUNCT
ejpam-5587	35	4	the	the	DET
ejpam-5587	35	5	minimum	minimum	ADJ
ejpam-5587	35	6	modulus	modulus	NOUN
ejpam-5587	35	7	,	,	PUNCT
ejpam-5587	35	8	m(t	m(t	NOUN
ejpam-5587	35	9	)	)	PUNCT
ejpam-5587	35	10	is	be	AUX
ejpam-5587	35	11	equal	equal	ADJ
ejpam-5587	35	12	to	to	ADP
ejpam-5587	35	13	the	the	DET
ejpam-5587	35	14	distance	distance	NOUN
ejpam-5587	35	15	of	of	ADP
ejpam-5587	35	16	0	0	NUM
ejpam-5587	35	17	from	from	ADP
ejpam-5587	35	18	spectrum	spectrum	NOUN
ejpam-5587	35	19	of	of	ADP
ejpam-5587	35	20	t	t	PROPN
ejpam-5587	35	21	.	.	PUNCT
ejpam-5587	36	1	(	(	PUNCT
ejpam-5587	36	2	iv	iv	X
ejpam-5587	36	3	)	)	PUNCT
ejpam-5587	36	4	t	t	PROPN
ejpam-5587	36	5	satisfies	satisfy	VERB
ejpam-5587	36	6	the	the	DET
ejpam-5587	36	7	weyl	weyl	PROPN
ejpam-5587	36	8	’s	’s	PART
ejpam-5587	36	9	theorem	theorem	NOUN
ejpam-5587	36	10	i.e.	i.e.	X
ejpam-5587	36	11	σ(t	σ(t	PROPN
ejpam-5587	36	12	)	)	PUNCT
ejpam-5587	37	1	\σw(t	\σw(t	ADV
ejpam-5587	37	2	)	)	PUNCT
ejpam-5587	37	3	=	=	SYM
ejpam-5587	37	4	π00(t	π00(t	ADJ
ejpam-5587	37	5	)	)	PUNCT
ejpam-5587	37	6	.	.	PUNCT
ejpam-5587	38	1	here	here	ADV
ejpam-5587	38	2	σw(t	σw(t	PUNCT
ejpam-5587	38	3	)	)	PUNCT
ejpam-5587	38	4	is	be	AUX
ejpam-5587	38	5	the	the	DET
ejpam-5587	38	6	weyl	weyl	PROPN
ejpam-5587	38	7	’s	’s	PART
ejpam-5587	38	8	spectrum	spectrum	NOUN
ejpam-5587	38	9	and	and	CCONJ
ejpam-5587	38	10	π00(t	π00(t	X
ejpam-5587	38	11	)	)	PUNCT
ejpam-5587	38	12	consists	consist	VERB
ejpam-5587	38	13	of	of	ADP
ejpam-5587	38	14	all	all	DET
ejpam-5587	38	15	isolated	isolated	ADJ
ejpam-5587	38	16	eigenvalues	eigenvalue	NOUN
ejpam-5587	38	17	of	of	ADP
ejpam-5587	38	18	t	t	PROPN
ejpam-5587	38	19	with	with	ADP
ejpam-5587	38	20	finite	finite	ADJ
ejpam-5587	38	21	multiplicity	multiplicity	NOUN
ejpam-5587	38	22	.	.	PUNCT
ejpam-5587	39	1	(	(	PUNCT
ejpam-5587	39	2	v	v	NOUN
ejpam-5587	39	3	)	)	PUNCT
ejpam-5587	39	4	if	if	SCONJ
ejpam-5587	39	5	µ	µ	NOUN
ejpam-5587	39	6	is	be	AUX
ejpam-5587	39	7	a	a	DET
ejpam-5587	39	8	non	non	ADJ
ejpam-5587	39	9	-	-	ADJ
ejpam-5587	39	10	zero	zero	ADJ
ejpam-5587	39	11	isolated	isolate	VERB
ejpam-5587	39	12	spectral	spectral	ADJ
ejpam-5587	39	13	value	value	NOUN
ejpam-5587	39	14	of	of	ADP
ejpam-5587	39	15	t	t	PROPN
ejpam-5587	39	16	,	,	PUNCT
ejpam-5587	39	17	then	then	ADV
ejpam-5587	39	18	the	the	DET
ejpam-5587	39	19	riesz	riesz	PROPN
ejpam-5587	39	20	projection	projection	NOUN
ejpam-5587	39	21	eµ	eµ	NOUN
ejpam-5587	39	22	with	with	ADP
ejpam-5587	39	23	respect	respect	NOUN
ejpam-5587	39	24	to	to	ADP
ejpam-5587	39	25	µ	µ	PROPN
ejpam-5587	39	26	is	be	AUX
ejpam-5587	39	27	self	self	NOUN
ejpam-5587	39	28	-	-	PUNCT
ejpam-5587	39	29	adjoint	adjoint	NOUN
ejpam-5587	39	30	and	and	CCONJ
ejpam-5587	39	31	satisfies	satisfie	NOUN
ejpam-5587	39	32	ran(eµ	ran(eµ	NOUN
ejpam-5587	39	33	)	)	PUNCT
ejpam-5587	40	1	=	=	SYM
ejpam-5587	40	2	ker(t	ker(t	NOUN
ejpam-5587	40	3	−	−	NUM
ejpam-5587	40	4	µi	µi	NOUN
ejpam-5587	40	5	)	)	PUNCT
ejpam-5587	40	6	=	=	PUNCT
ejpam-5587	40	7	ker(t	ker(t	NOUN
ejpam-5587	40	8	−	−	NOUN
ejpam-5587	40	9	µi)∗.	µi)∗.	ADV
ejpam-5587	40	10	the	the	DET
ejpam-5587	40	11	study	study	NOUN
ejpam-5587	40	12	of	of	ADP
ejpam-5587	40	13	weyl	weyl	PROPN
ejpam-5587	40	14	’s	’s	PART
ejpam-5587	40	15	theorem	theorem	NOUN
ejpam-5587	40	16	and	and	CCONJ
ejpam-5587	40	17	the	the	DET
ejpam-5587	40	18	self	self	NOUN
ejpam-5587	40	19	-	-	PUNCT
ejpam-5587	40	20	adjointness	adjointness	NOUN
ejpam-5587	40	21	of	of	ADP
ejpam-5587	40	22	the	the	DET
ejpam-5587	40	23	riesz	riesz	NOUN
ejpam-5587	40	24	projection	projection	NOUN
ejpam-5587	40	25	for	for	ADP
ejpam-5587	40	26	isolated	isolated	ADJ
ejpam-5587	40	27	spectral	spectral	ADJ
ejpam-5587	40	28	values	value	NOUN
ejpam-5587	40	29	has	have	AUX
ejpam-5587	40	30	been	be	AUX
ejpam-5587	40	31	explored	explore	VERB
ejpam-5587	40	32	for	for	ADP
ejpam-5587	40	33	various	various	ADJ
ejpam-5587	40	34	operator	operator	NOUN
ejpam-5587	40	35	classes	class	NOUN
ejpam-5587	40	36	.	.	PUNCT
ejpam-5587	41	1	coburn	coburn	NOUN
ejpam-5587	42	1	[	[	X
ejpam-5587	42	2	5	5	NUM
ejpam-5587	42	3	]	]	PUNCT
ejpam-5587	42	4	established	establish	VERB
ejpam-5587	42	5	these	these	DET
ejpam-5587	42	6	properties	property	NOUN
ejpam-5587	42	7	for	for	ADP
ejpam-5587	42	8	certain	certain	ADJ
ejpam-5587	42	9	non	non	ADJ
ejpam-5587	42	10	-	-	ADJ
ejpam-5587	42	11	normal	normal	ADJ
ejpam-5587	42	12	operators	operator	NOUN
ejpam-5587	42	13	,	,	PUNCT
ejpam-5587	42	14	including	include	VERB
ejpam-5587	42	15	hyponormal	hyponormal	ADJ
ejpam-5587	42	16	and	and	CCONJ
ejpam-5587	42	17	toeplitz	toeplitz	NOUN
ejpam-5587	42	18	operators	operator	NOUN
ejpam-5587	42	19	.	.	PUNCT
ejpam-5587	43	1	schmoeger	schmoeger	NOUN
ejpam-5587	44	1	[	[	X
ejpam-5587	44	2	22	22	NUM
ejpam-5587	44	3	]	]	PUNCT
ejpam-5587	44	4	expanded	expand	VERB
ejpam-5587	44	5	this	this	DET
ejpam-5587	44	6	work	work	NOUN
ejpam-5587	44	7	to	to	PART
ejpam-5587	44	8	bounded	bound	VERB
ejpam-5587	44	9	totally	totally	ADV
ejpam-5587	44	10	paranormal	paranormal	ADJ
ejpam-5587	44	11	operators	operator	NOUN
ejpam-5587	44	12	,	,	PUNCT
ejpam-5587	44	13	drawing	draw	VERB
ejpam-5587	44	14	on	on	ADP
ejpam-5587	44	15	ando	ando	PROPN
ejpam-5587	44	16	’s	’s	PART
ejpam-5587	44	17	characterization	characterization	NOUN
ejpam-5587	44	18	[	[	X
ejpam-5587	44	19	3	3	X
ejpam-5587	44	20	]	]	PUNCT
ejpam-5587	44	21	of	of	ADP
ejpam-5587	44	22	paranormal	paranormal	ADJ
ejpam-5587	44	23	operators	operator	NOUN
ejpam-5587	44	24	.	.	PUNCT
ejpam-5587	45	1	since	since	SCONJ
ejpam-5587	45	2	ando	ando	PROPN
ejpam-5587	45	3	’s	’s	PART
ejpam-5587	45	4	characterization	characterization	NOUN
ejpam-5587	45	5	does	do	AUX
ejpam-5587	45	6	not	not	PART
ejpam-5587	45	7	extend	extend	VERB
ejpam-5587	45	8	to	to	ADP
ejpam-5587	45	9	unbounded	unbounded	ADJ
ejpam-5587	45	10	paranormal	paranormal	ADJ
ejpam-5587	45	11	operators	operator	NOUN
ejpam-5587	45	12	,	,	PUNCT
ejpam-5587	45	13	and	and	CCONJ
ejpam-5587	45	14	methods	method	NOUN
ejpam-5587	45	15	for	for	ADP
ejpam-5587	45	16	bounded	bounded	ADJ
ejpam-5587	45	17	operators	operator	NOUN
ejpam-5587	45	18	are	be	AUX
ejpam-5587	45	19	unsuitable	unsuitable	ADJ
ejpam-5587	45	20	here	here	ADV
ejpam-5587	45	21	,	,	PUNCT
ejpam-5587	45	22	we	we	PRON
ejpam-5587	45	23	aim	aim	VERB
ejpam-5587	45	24	to	to	PART
ejpam-5587	45	25	establish	establish	VERB
ejpam-5587	45	26	properties	property	NOUN
ejpam-5587	45	27	(	(	PUNCT
ejpam-5587	45	28	iv	iv	X
ejpam-5587	45	29	)	)	PUNCT
ejpam-5587	45	30	and	and	CCONJ
ejpam-5587	45	31	(	(	PUNCT
ejpam-5587	45	32	v	v	NOUN
ejpam-5587	45	33	)	)	PUNCT
ejpam-5587	45	34	using	use	VERB
ejpam-5587	45	35	an	an	DET
ejpam-5587	45	36	alternative	alternative	ADJ
ejpam-5587	45	37	approach	approach	NOUN
ejpam-5587	45	38	.	.	PUNCT
ejpam-5587	46	1	gupta	gupta	NOUN
ejpam-5587	46	2	and	and	CCONJ
ejpam-5587	46	3	mamtani	mamtani	NOUN
ejpam-5587	47	1	[	[	X
ejpam-5587	47	2	11	11	NUM
ejpam-5587	47	3	]	]	PUNCT
ejpam-5587	47	4	showed	show	VERB
ejpam-5587	47	5	that	that	SCONJ
ejpam-5587	47	6	closed	close	VERB
ejpam-5587	47	7	hyponormal	hyponormal	ADJ
ejpam-5587	47	8	operators	operator	NOUN
ejpam-5587	47	9	satisfy	satisfy	VERB
ejpam-5587	47	10	weyl	weyl	PROPN
ejpam-5587	47	11	’s	’s	PART
ejpam-5587	47	12	theorem	theorem	PROPN
ejpam-5587	47	13	.	.	PUNCT
ejpam-5587	48	1	in	in	ADP
ejpam-5587	48	2	a	a	DET
ejpam-5587	48	3	follow	follow	VERB
ejpam-5587	48	4	-	-	PUNCT
ejpam-5587	48	5	up	up	ADP
ejpam-5587	48	6	study	study	NOUN
ejpam-5587	48	7	[	[	X
ejpam-5587	48	8	10	10	NUM
ejpam-5587	48	9	]	]	PUNCT
ejpam-5587	48	10	,	,	PUNCT
ejpam-5587	48	11	they	they	PRON
ejpam-5587	48	12	outlined	outline	VERB
ejpam-5587	48	13	key	key	ADJ
ejpam-5587	48	14	conditions	condition	NOUN
ejpam-5587	48	15	required	require	VERB
ejpam-5587	48	16	for	for	ADP
ejpam-5587	48	17	the	the	DET
ejpam-5587	48	18	orthogonal	orthogonal	ADJ
ejpam-5587	48	19	direct	direct	ADJ
ejpam-5587	48	20	sum	sum	NOUN
ejpam-5587	48	21	of	of	ADP
ejpam-5587	48	22	densely	densely	ADV
ejpam-5587	48	23	defined	define	VERB
ejpam-5587	48	24	closed	closed	ADJ
ejpam-5587	48	25	operators	operator	NOUN
ejpam-5587	48	26	to	to	PART
ejpam-5587	48	27	fulfill	fulfill	VERB
ejpam-5587	48	28	weyl	weyl	PROPN
ejpam-5587	48	29	’s	’s	PART
ejpam-5587	48	30	theorem	theorem	PROPN
ejpam-5587	48	31	.	.	PUNCT
ejpam-5587	49	1	the	the	DET
ejpam-5587	49	2	paper	paper	NOUN
ejpam-5587	49	3	is	be	AUX
ejpam-5587	49	4	organized	organize	VERB
ejpam-5587	49	5	into	into	ADP
ejpam-5587	49	6	four	four	NUM
ejpam-5587	49	7	sections	section	NOUN
ejpam-5587	49	8	for	for	ADP
ejpam-5587	49	9	clarity	clarity	NOUN
ejpam-5587	49	10	.	.	PUNCT
ejpam-5587	50	1	in	in	ADP
ejpam-5587	50	2	section	section	NOUN
ejpam-5587	50	3	2	2	NUM
ejpam-5587	50	4	,	,	PUNCT
ejpam-5587	50	5	we	we	PRON
ejpam-5587	50	6	introduce	introduce	VERB
ejpam-5587	50	7	key	key	ADJ
ejpam-5587	50	8	notations	notation	NOUN
ejpam-5587	50	9	and	and	CCONJ
ejpam-5587	50	10	summarize	summarize	VERB
ejpam-5587	50	11	relevant	relevant	ADJ
ejpam-5587	50	12	established	establish	VERB
ejpam-5587	50	13	results	result	NOUN
ejpam-5587	50	14	that	that	PRON
ejpam-5587	50	15	will	will	AUX
ejpam-5587	50	16	be	be	AUX
ejpam-5587	50	17	used	use	VERB
ejpam-5587	50	18	throughout	throughout	ADP
ejpam-5587	50	19	the	the	DET
ejpam-5587	50	20	study	study	NOUN
ejpam-5587	50	21	.	.	PUNCT
ejpam-5587	51	1	s.	s.	PROPN
ejpam-5587	51	2	alnabulsi	alnabulsi	PROPN
ejpam-5587	51	3	,	,	PUNCT
ejpam-5587	51	4	m.h.m	m.h.m	PROPN
ejpam-5587	51	5	.	.	PUNCT
ejpam-5587	51	6	rashid	rashid	PROPN
ejpam-5587	51	7	/	/	SYM
ejpam-5587	51	8	eur	eur	PROPN
ejpam-5587	51	9	.	.	PUNCT
ejpam-5587	52	1	j.	j.	PROPN
ejpam-5587	52	2	pure	pure	PROPN
ejpam-5587	52	3	appl	appl	PROPN
ejpam-5587	52	4	.	.	PROPN
ejpam-5587	52	5	math	math	PROPN
ejpam-5587	52	6	,	,	PUNCT
ejpam-5587	52	7	18	18	NUM
ejpam-5587	52	8	(	(	PUNCT
ejpam-5587	52	9	1	1	NUM
ejpam-5587	52	10	)	)	PUNCT
ejpam-5587	52	11	(	(	PUNCT
ejpam-5587	52	12	2025	2025	NUM
ejpam-5587	52	13	)	)	PUNCT
ejpam-5587	52	14	,	,	PUNCT
ejpam-5587	52	15	5587	5587	NUM
ejpam-5587	52	16	3	3	NUM
ejpam-5587	52	17	of	of	ADP
ejpam-5587	52	18	20	20	NUM
ejpam-5587	52	19	section	section	NOUN
ejpam-5587	52	20	3	3	NUM
ejpam-5587	52	21	focuses	focus	VERB
ejpam-5587	52	22	on	on	ADP
ejpam-5587	52	23	examining	examine	VERB
ejpam-5587	52	24	spectral	spectral	ADJ
ejpam-5587	52	25	properties	property	NOUN
ejpam-5587	52	26	associated	associate	VERB
ejpam-5587	52	27	with	with	ADP
ejpam-5587	52	28	densely	densely	ADV
ejpam-5587	52	29	defined	define	VERB
ejpam-5587	52	30	closed	close	VERB
ejpam-5587	52	31	totally	totally	ADV
ejpam-5587	52	32	paranormal	paranormal	ADJ
ejpam-5587	52	33	operators	operator	NOUN
ejpam-5587	52	34	.	.	PUNCT
ejpam-5587	53	1	finally	finally	ADV
ejpam-5587	53	2	,	,	PUNCT
ejpam-5587	53	3	section	section	NOUN
ejpam-5587	53	4	4	4	NUM
ejpam-5587	53	5	presents	present	VERB
ejpam-5587	53	6	the	the	DET
ejpam-5587	53	7	proof	proof	NOUN
ejpam-5587	53	8	of	of	ADP
ejpam-5587	53	9	weyl	weyl	PROPN
ejpam-5587	53	10	’s	’s	PART
ejpam-5587	53	11	theorem	theorem	NOUN
ejpam-5587	53	12	for	for	ADP
ejpam-5587	53	13	these	these	DET
ejpam-5587	53	14	operators	operator	NOUN
ejpam-5587	53	15	.	.	PUNCT
ejpam-5587	54	1	2	2	X
ejpam-5587	54	2	.	.	NUM
ejpam-5587	54	3	notations	notation	NOUN
ejpam-5587	54	4	and	and	CCONJ
ejpam-5587	54	5	preliminaries	preliminary	NOUN
ejpam-5587	54	6	in	in	ADP
ejpam-5587	54	7	this	this	DET
ejpam-5587	54	8	article	article	NOUN
ejpam-5587	54	9	,	,	PUNCT
ejpam-5587	54	10	we	we	PRON
ejpam-5587	54	11	explore	explore	VERB
ejpam-5587	54	12	intricate	intricate	ADJ
ejpam-5587	54	13	hilbert	hilbert	NOUN
ejpam-5587	54	14	spaces	space	NOUN
ejpam-5587	54	15	,	,	PUNCT
ejpam-5587	54	16	represented	represent	VERB
ejpam-5587	54	17	as	as	ADP
ejpam-5587	54	18	h	h	NOUN
ejpam-5587	54	19	,	,	PUNCT
ejpam-5587	54	20	h1,h2	h1,h2	PROPN
ejpam-5587	54	21	,	,	PUNCT
ejpam-5587	54	22	and	and	CCONJ
ejpam-5587	54	23	so	so	ADV
ejpam-5587	54	24	forth	forth	ADV
ejpam-5587	54	25	.	.	PUNCT
ejpam-5587	55	1	the	the	DET
ejpam-5587	55	2	inner	inner	ADJ
ejpam-5587	55	3	product	product	NOUN
ejpam-5587	55	4	and	and	CCONJ
ejpam-5587	55	5	the	the	DET
ejpam-5587	55	6	corresponding	corresponding	ADJ
ejpam-5587	55	7	norm	norm	NOUN
ejpam-5587	55	8	are	be	AUX
ejpam-5587	55	9	symbolized	symbolize	VERB
ejpam-5587	55	10	by	by	ADP
ejpam-5587	55	11	⟨	⟨	NOUN
ejpam-5587	55	12	·	·	NUM
ejpam-5587	55	13	,	,	PUNCT
ejpam-5587	55	14	·	·	PUNCT
ejpam-5587	55	15	⟩	⟩	NOUN
ejpam-5587	55	16	and	and	CCONJ
ejpam-5587	55	17	∥·∥	∥·∥	PROPN
ejpam-5587	55	18	,	,	PUNCT
ejpam-5587	55	19	respectively	respectively	ADV
ejpam-5587	55	20	.	.	PUNCT
ejpam-5587	56	1	the	the	DET
ejpam-5587	56	2	set	set	NOUN
ejpam-5587	56	3	of	of	ADP
ejpam-5587	56	4	all	all	DET
ejpam-5587	56	5	linear	linear	ADJ
ejpam-5587	56	6	operators	operator	NOUN
ejpam-5587	56	7	on	on	ADP
ejpam-5587	56	8	h	h	NOUN
ejpam-5587	56	9	is	be	AUX
ejpam-5587	56	10	denoted	denote	VERB
ejpam-5587	56	11	as	as	ADP
ejpam-5587	56	12	l(h	l(h	PROPN
ejpam-5587	56	13	)	)	PUNCT
ejpam-5587	56	14	,	,	PUNCT
ejpam-5587	56	15	while	while	SCONJ
ejpam-5587	56	16	the	the	DET
ejpam-5587	56	17	collection	collection	NOUN
ejpam-5587	56	18	of	of	ADP
ejpam-5587	56	19	all	all	DET
ejpam-5587	56	20	bounded	bound	VERB
ejpam-5587	56	21	linear	linear	PROPN
ejpam-5587	56	22	operators	operator	NOUN
ejpam-5587	56	23	is	be	AUX
ejpam-5587	56	24	represented	represent	VERB
ejpam-5587	56	25	as	as	ADP
ejpam-5587	56	26	b(h	b(h	NOUN
ejpam-5587	56	27	)	)	PUNCT
ejpam-5587	56	28	.	.	PUNCT
ejpam-5587	57	1	for	for	ADP
ejpam-5587	57	2	a	a	DET
ejpam-5587	57	3	linear	linear	ADJ
ejpam-5587	57	4	operator	operator	NOUN
ejpam-5587	57	5	t	t	PROPN
ejpam-5587	57	6	∈	∈	PROPN
ejpam-5587	57	7	l(h	l(h	PROPN
ejpam-5587	57	8	)	)	PUNCT
ejpam-5587	57	9	,	,	PUNCT
ejpam-5587	57	10	we	we	PRON
ejpam-5587	57	11	use	use	VERB
ejpam-5587	57	12	d(t	d(t	PROPN
ejpam-5587	57	13	)	)	PUNCT
ejpam-5587	57	14	,	,	PUNCT
ejpam-5587	57	15	ker(t	ker(t	NOUN
ejpam-5587	57	16	)	)	PUNCT
ejpam-5587	57	17	,	,	PUNCT
ejpam-5587	57	18	and	and	CCONJ
ejpam-5587	57	19	ran(t	ran(t	NUM
ejpam-5587	57	20	)	)	PUNCT
ejpam-5587	57	21	to	to	PART
ejpam-5587	57	22	signify	signify	VERB
ejpam-5587	57	23	its	its	PRON
ejpam-5587	57	24	domain	domain	NOUN
ejpam-5587	57	25	,	,	PUNCT
ejpam-5587	57	26	null	null	ADJ
ejpam-5587	57	27	space	space	NOUN
ejpam-5587	57	28	,	,	PUNCT
ejpam-5587	57	29	and	and	CCONJ
ejpam-5587	57	30	range	range	NOUN
ejpam-5587	57	31	space	space	NOUN
ejpam-5587	57	32	,	,	PUNCT
ejpam-5587	57	33	respectively	respectively	ADV
ejpam-5587	57	34	.	.	PUNCT
ejpam-5587	58	1	a	a	DET
ejpam-5587	58	2	linear	linear	ADJ
ejpam-5587	58	3	operator	operator	NOUN
ejpam-5587	58	4	t	t	NOUN
ejpam-5587	58	5	is	be	AUX
ejpam-5587	58	6	termed	term	VERB
ejpam-5587	58	7	a	a	DET
ejpam-5587	58	8	densely	densely	ADV
ejpam-5587	58	9	defined	define	VERB
ejpam-5587	58	10	operator	operator	NOUN
ejpam-5587	58	11	if	if	SCONJ
ejpam-5587	58	12	d(t	d(t	PROPN
ejpam-5587	58	13	)	)	PUNCT
ejpam-5587	59	1	=	=	PUNCT
ejpam-5587	60	1	h.	h.	NOUN
ejpam-5587	61	1	if	if	SCONJ
ejpam-5587	61	2	t	t	PROPN
ejpam-5587	61	3	∈	∈	PROPN
ejpam-5587	61	4	l(h	l(h	PROPN
ejpam-5587	61	5	)	)	PUNCT
ejpam-5587	61	6	and	and	CCONJ
ejpam-5587	61	7	m	m	PROPN
ejpam-5587	61	8	is	be	AUX
ejpam-5587	61	9	a	a	DET
ejpam-5587	61	10	closed	closed	ADJ
ejpam-5587	61	11	subspace	subspace	NOUN
ejpam-5587	61	12	of	of	ADP
ejpam-5587	61	13	h	h	NOUN
ejpam-5587	61	14	,	,	PUNCT
ejpam-5587	61	15	then	then	ADV
ejpam-5587	61	16	m	m	VERB
ejpam-5587	61	17	is	be	AUX
ejpam-5587	61	18	said	say	VERB
ejpam-5587	61	19	to	to	PART
ejpam-5587	61	20	be	be	AUX
ejpam-5587	61	21	invariant	invariant	ADJ
ejpam-5587	61	22	under	under	ADP
ejpam-5587	61	23	t	t	PROPN
ejpam-5587	61	24	,	,	PUNCT
ejpam-5587	61	25	if	if	SCONJ
ejpam-5587	61	26	for	for	ADP
ejpam-5587	61	27	every	every	DET
ejpam-5587	61	28	x	x	PROPN
ejpam-5587	61	29	∈	∈	PROPN
ejpam-5587	61	30	d(t	d(t	PROPN
ejpam-5587	61	31	)	)	PUNCT
ejpam-5587	62	1	∩m	∩m	PROPN
ejpam-5587	62	2	,	,	PUNCT
ejpam-5587	62	3	tx	tx	PROPN
ejpam-5587	62	4	is	be	AUX
ejpam-5587	62	5	in	in	ADP
ejpam-5587	62	6	m.	m.	NOUN
ejpam-5587	62	7	we	we	PRON
ejpam-5587	62	8	denote	denote	VERB
ejpam-5587	62	9	the	the	DET
ejpam-5587	62	10	identity	identity	NOUN
ejpam-5587	62	11	operator	operator	NOUN
ejpam-5587	62	12	on	on	ADP
ejpam-5587	62	13	m	m	PROPN
ejpam-5587	62	14	by	by	ADP
ejpam-5587	62	15	i	i	PROPN
ejpam-5587	62	16	m	m	PROPN
ejpam-5587	62	17	,	,	PUNCT
ejpam-5587	62	18	the	the	DET
ejpam-5587	62	19	orthogonal	orthogonal	ADJ
ejpam-5587	62	20	projection	projection	NOUN
ejpam-5587	62	21	on	on	ADP
ejpam-5587	62	22	m	m	PROPN
ejpam-5587	62	23	by	by	ADP
ejpam-5587	62	24	pm	pm	NOUN
ejpam-5587	62	25	.	.	PUNCT
ejpam-5587	63	1	the	the	DET
ejpam-5587	63	2	unit	unit	NOUN
ejpam-5587	63	3	sphere	sphere	ADV
ejpam-5587	63	4	of	of	ADP
ejpam-5587	63	5	m	m	PROPN
ejpam-5587	63	6	is	be	AUX
ejpam-5587	63	7	tm	tm	NOUN
ejpam-5587	63	8	:	:	PUNCT
ejpam-5587	63	9	=	=	SYM
ejpam-5587	63	10	{	{	PUNCT
ejpam-5587	63	11	x	x	SYM
ejpam-5587	63	12	∈	∈	NOUN
ejpam-5587	63	13	m	m	VERB
ejpam-5587	63	14	:	:	PUNCT
ejpam-5587	63	15	∥x∥	∥x∥	NOUN
ejpam-5587	63	16	=	=	NOUN
ejpam-5587	63	17	1	1	NUM
ejpam-5587	63	18	}	}	PUNCT
ejpam-5587	63	19	.	.	PUNCT
ejpam-5587	64	1	the	the	DET
ejpam-5587	64	2	restriction	restriction	NOUN
ejpam-5587	64	3	of	of	ADP
ejpam-5587	64	4	t	t	PROPN
ejpam-5587	64	5	to	to	ADP
ejpam-5587	64	6	m	m	PROPN
ejpam-5587	64	7	is	be	AUX
ejpam-5587	64	8	an	an	DET
ejpam-5587	64	9	operator	operator	NOUN
ejpam-5587	64	10	t	t	NOUN
ejpam-5587	64	11	|m	|m	NOUN
ejpam-5587	64	12	:	:	PUNCT
ejpam-5587	64	13	m	m	VERB
ejpam-5587	64	14	∩d(t	∩d(t	ADJ
ejpam-5587	64	15	)	)	PUNCT
ejpam-5587	64	16	→	→	SYM
ejpam-5587	64	17	h	h	PRON
ejpam-5587	64	18	defined	define	VERB
ejpam-5587	64	19	by	by	ADP
ejpam-5587	64	20	t	t	PROPN
ejpam-5587	64	21	|mx	|mx	NOUN
ejpam-5587	64	22	=	=	SYM
ejpam-5587	64	23	tx	tx	PROPN
ejpam-5587	64	24	,	,	PUNCT
ejpam-5587	64	25	for	for	ADP
ejpam-5587	64	26	all	all	DET
ejpam-5587	64	27	x	x	SYM
ejpam-5587	64	28	∈	∈	PROPN
ejpam-5587	64	29	m∩d(t	m∩d(t	NOUN
ejpam-5587	64	30	)	)	PUNCT
ejpam-5587	64	31	.	.	PUNCT
ejpam-5587	65	1	if	if	SCONJ
ejpam-5587	65	2	m	m	NOUN
ejpam-5587	65	3	is	be	AUX
ejpam-5587	65	4	invariant	invariant	ADJ
ejpam-5587	65	5	under	under	ADP
ejpam-5587	65	6	t	t	PROPN
ejpam-5587	65	7	,	,	PUNCT
ejpam-5587	65	8	then	then	ADV
ejpam-5587	65	9	t	t	PROPN
ejpam-5587	65	10	|m	|m	NOUN
ejpam-5587	65	11	is	be	AUX
ejpam-5587	65	12	an	an	DET
ejpam-5587	65	13	operator	operator	NOUN
ejpam-5587	65	14	from	from	ADP
ejpam-5587	65	15	d(t	d(t	PROPN
ejpam-5587	65	16	)	)	PUNCT
ejpam-5587	65	17	∩m	∩m	PROPN
ejpam-5587	65	18	into	into	ADP
ejpam-5587	65	19	m.	m.	NOUN
ejpam-5587	65	20	an	an	DET
ejpam-5587	65	21	operator	operator	NOUN
ejpam-5587	65	22	t	t	PROPN
ejpam-5587	65	23	∈	∈	PROPN
ejpam-5587	65	24	l(h	l(h	PROPN
ejpam-5587	65	25	)	)	PUNCT
ejpam-5587	65	26	is	be	AUX
ejpam-5587	65	27	said	say	VERB
ejpam-5587	65	28	to	to	PART
ejpam-5587	65	29	be	be	AUX
ejpam-5587	65	30	closed	close	VERB
ejpam-5587	65	31	if	if	SCONJ
ejpam-5587	65	32	for	for	ADP
ejpam-5587	65	33	any	any	DET
ejpam-5587	65	34	sequence	sequence	NOUN
ejpam-5587	65	35	{	{	PUNCT
ejpam-5587	65	36	xn	xn	NOUN
ejpam-5587	65	37	}	}	PUNCT
ejpam-5587	65	38	⊆	⊆	NUM
ejpam-5587	65	39	d(t	d(t	PROPN
ejpam-5587	65	40	)	)	PUNCT
ejpam-5587	65	41	with	with	ADP
ejpam-5587	65	42	xn	xn	PROPN
ejpam-5587	65	43	→	→	SYM
ejpam-5587	65	44	x	x	X
ejpam-5587	65	45	and	and	CCONJ
ejpam-5587	65	46	txn	txn	X
ejpam-5587	65	47	→	→	SYM
ejpam-5587	65	48	y	y	PROPN
ejpam-5587	65	49	then	then	ADV
ejpam-5587	65	50	x	x	PROPN
ejpam-5587	65	51	∈	∈	PROPN
ejpam-5587	65	52	d(t	d(t	PROPN
ejpam-5587	65	53	)	)	PUNCT
ejpam-5587	65	54	and	and	CCONJ
ejpam-5587	65	55	tx	tx	X
ejpam-5587	65	56	=	=	PUNCT
ejpam-5587	65	57	y.	y.	NOUN
ejpam-5587	65	58	in	in	ADP
ejpam-5587	65	59	this	this	DET
ejpam-5587	65	60	document	document	NOUN
ejpam-5587	65	61	,	,	PUNCT
ejpam-5587	65	62	the	the	DET
ejpam-5587	65	63	notation	notation	NOUN
ejpam-5587	65	64	c(h1,h2	c(h1,h2	NOUN
ejpam-5587	65	65	)	)	PUNCT
ejpam-5587	65	66	will	will	AUX
ejpam-5587	65	67	be	be	AUX
ejpam-5587	65	68	employed	employ	VERB
ejpam-5587	65	69	to	to	PART
ejpam-5587	65	70	denote	denote	VERB
ejpam-5587	65	71	the	the	DET
ejpam-5587	65	72	collection	collection	NOUN
ejpam-5587	65	73	of	of	ADP
ejpam-5587	65	74	closed	closed	ADJ
ejpam-5587	65	75	linear	linear	PROPN
ejpam-5587	65	76	operators	operator	NOUN
ejpam-5587	65	77	such	such	ADJ
ejpam-5587	65	78	that	that	DET
ejpam-5587	65	79	d(t	d(t	PROPN
ejpam-5587	65	80	)	)	PUNCT
ejpam-5587	65	81	⊆	⊆	NUM
ejpam-5587	65	82	h1	h1	NOUN
ejpam-5587	65	83	and	and	CCONJ
ejpam-5587	65	84	ran(t	ran(t	PROPN
ejpam-5587	65	85	)	)	PUNCT
ejpam-5587	65	86	⊆	⊆	NUM
ejpam-5587	65	87	h2	h2	NOUN
ejpam-5587	65	88	.	.	PUNCT
ejpam-5587	66	1	in	in	ADP
ejpam-5587	66	2	the	the	DET
ejpam-5587	66	3	case	case	NOUN
ejpam-5587	66	4	where	where	SCONJ
ejpam-5587	66	5	h1	h1	PROPN
ejpam-5587	66	6	equals	equal	VERB
ejpam-5587	66	7	h2	h2	NOUN
ejpam-5587	66	8	,	,	PUNCT
ejpam-5587	66	9	we	we	PRON
ejpam-5587	66	10	will	will	AUX
ejpam-5587	66	11	use	use	VERB
ejpam-5587	66	12	the	the	DET
ejpam-5587	66	13	shorthand	shorthand	NOUN
ejpam-5587	66	14	c(h	c(h	VERB
ejpam-5587	66	15	)	)	PUNCT
ejpam-5587	66	16	.	.	PUNCT
ejpam-5587	67	1	it	it	PRON
ejpam-5587	67	2	is	be	AUX
ejpam-5587	67	3	known	know	VERB
ejpam-5587	67	4	that	that	SCONJ
ejpam-5587	67	5	every	every	DET
ejpam-5587	67	6	densely	densely	ADV
ejpam-5587	67	7	defined	define	VERB
ejpam-5587	67	8	operator	operator	NOUN
ejpam-5587	67	9	t	t	PROPN
ejpam-5587	67	10	∈	∈	PROPN
ejpam-5587	67	11	c(h1,h2	c(h1,h2	PROPN
ejpam-5587	67	12	)	)	PUNCT
ejpam-5587	67	13	has	have	VERB
ejpam-5587	67	14	a	a	DET
ejpam-5587	67	15	unique	unique	ADJ
ejpam-5587	67	16	adjoint	adjoint	NOUN
ejpam-5587	67	17	in	in	ADP
ejpam-5587	67	18	c(h2,h1	c(h2,h1	NOUN
ejpam-5587	67	19	)	)	PUNCT
ejpam-5587	67	20	,	,	PUNCT
ejpam-5587	67	21	that	that	ADV
ejpam-5587	67	22	is	is	ADV
ejpam-5587	67	23	,	,	PUNCT
ejpam-5587	67	24	there	there	PRON
ejpam-5587	67	25	exists	exist	VERB
ejpam-5587	67	26	a	a	DET
ejpam-5587	67	27	unique	unique	ADJ
ejpam-5587	67	28	t	t	NOUN
ejpam-5587	67	29	∗	∗	NOUN
ejpam-5587	67	30	∈	∈	PROPN
ejpam-5587	67	31	c(h2,h1	c(h2,h1	NOUN
ejpam-5587	67	32	)	)	PUNCT
ejpam-5587	67	33	such	such	ADJ
ejpam-5587	67	34	that	that	PRON
ejpam-5587	67	35	⟨tx	⟨tx	PROPN
ejpam-5587	67	36	,	,	PUNCT
ejpam-5587	67	37	y⟩	y⟩	NOUN
ejpam-5587	67	38	=	=	PUNCT
ejpam-5587	67	39	⟨x	⟨x	NUM
ejpam-5587	67	40	,	,	PUNCT
ejpam-5587	67	41	t	t	PROPN
ejpam-5587	67	42	∗y⟩	∗y⟩	PROPN
ejpam-5587	67	43	for	for	ADP
ejpam-5587	67	44	all	all	DET
ejpam-5587	67	45	x	x	SYM
ejpam-5587	67	46	∈	∈	PROPN
ejpam-5587	67	47	d(t	d(t	PROPN
ejpam-5587	67	48	)	)	PUNCT
ejpam-5587	67	49	and	and	CCONJ
ejpam-5587	67	50	y	y	PROPN
ejpam-5587	67	51	∈	∈	PROPN
ejpam-5587	67	52	d(t	d(t	PROPN
ejpam-5587	67	53	∗	∗	NOUN
ejpam-5587	67	54	)	)	PUNCT
ejpam-5587	67	55	.	.	PUNCT
ejpam-5587	68	1	remark	remark	PROPN
ejpam-5587	68	2	1	1	NUM
ejpam-5587	68	3	.	.	PUNCT
ejpam-5587	69	1	by	by	ADP
ejpam-5587	69	2	the	the	DET
ejpam-5587	69	3	closed	closed	ADJ
ejpam-5587	69	4	graph	graph	NOUN
ejpam-5587	69	5	theorem	theorem	NOUN
ejpam-5587	69	6	(	(	PUNCT
ejpam-5587	69	7	cf	cf	NOUN
ejpam-5587	69	8	.	.	PUNCT
ejpam-5587	70	1	[	[	X
ejpam-5587	70	2	18	18	NUM
ejpam-5587	70	3	,	,	PUNCT
ejpam-5587	70	4	theorem	theorem	VERB
ejpam-5587	70	5	7.1	7.1	NUM
ejpam-5587	70	6	,	,	PUNCT
ejpam-5587	70	7	page	page	NOUN
ejpam-5587	70	8	231	231	NUM
ejpam-5587	70	9	]	]	PUNCT
ejpam-5587	70	10	,	,	PUNCT
ejpam-5587	70	11	it	it	PRON
ejpam-5587	70	12	follows	follow	VERB
ejpam-5587	70	13	that	that	SCONJ
ejpam-5587	70	14	a	a	DET
ejpam-5587	70	15	closed	closed	ADJ
ejpam-5587	70	16	operator	operator	NOUN
ejpam-5587	70	17	t	t	PROPN
ejpam-5587	70	18	∈	∈	PROPN
ejpam-5587	70	19	c(h1,h2	c(h1,h2	PROPN
ejpam-5587	70	20	)	)	PUNCT
ejpam-5587	70	21	with	with	ADP
ejpam-5587	70	22	d(t	d(t	PROPN
ejpam-5587	70	23	)	)	PUNCT
ejpam-5587	71	1	=	=	PUNCT
ejpam-5587	71	2	h1	h1	PROPN
ejpam-5587	71	3	is	be	AUX
ejpam-5587	71	4	bounded	bound	VERB
ejpam-5587	71	5	.	.	PUNCT
ejpam-5587	72	1	lemma	lemma	PROPN
ejpam-5587	72	2	1	1	NUM
ejpam-5587	72	3	.	.	PUNCT
ejpam-5587	73	1	[	[	X
ejpam-5587	73	2	9	9	NUM
ejpam-5587	73	3	]	]	PUNCT
ejpam-5587	73	4	let	let	VERB
ejpam-5587	73	5	t	t	PROPN
ejpam-5587	73	6	∈	∈	PROPN
ejpam-5587	73	7	l(h	l(h	PROPN
ejpam-5587	73	8	)	)	PUNCT
ejpam-5587	73	9	be	be	AUX
ejpam-5587	73	10	a	a	DET
ejpam-5587	73	11	densely	densely	ADV
ejpam-5587	73	12	defined	define	VERB
ejpam-5587	73	13	closed	closed	ADJ
ejpam-5587	73	14	operator	operator	NOUN
ejpam-5587	73	15	.	.	PUNCT
ejpam-5587	74	1	then	then	ADV
ejpam-5587	74	2	d(t	d(t	PROPN
ejpam-5587	74	3	)	)	PUNCT
ejpam-5587	74	4	∩	∩	ADJ
ejpam-5587	74	5	ker(t	ker(t	NOUN
ejpam-5587	74	6	)	)	PUNCT
ejpam-5587	74	7	⊥	⊥	NOUN
ejpam-5587	74	8	=	=	SYM
ejpam-5587	74	9	ker(t	ker(t	NOUN
ejpam-5587	74	10	)	)	PUNCT
ejpam-5587	74	11	⊥.	⊥.	PROPN
ejpam-5587	74	12	if	if	SCONJ
ejpam-5587	74	13	s	s	NOUN
ejpam-5587	74	14	and	and	CCONJ
ejpam-5587	74	15	t	t	PROPN
ejpam-5587	74	16	are	be	AUX
ejpam-5587	74	17	two	two	NUM
ejpam-5587	74	18	closed	closed	ADJ
ejpam-5587	74	19	operators	operator	NOUN
ejpam-5587	74	20	,	,	PUNCT
ejpam-5587	74	21	then	then	ADV
ejpam-5587	74	22	s	s	VERB
ejpam-5587	74	23	is	be	AUX
ejpam-5587	74	24	called	call	VERB
ejpam-5587	74	25	an	an	DET
ejpam-5587	74	26	extension	extension	NOUN
ejpam-5587	74	27	of	of	ADP
ejpam-5587	74	28	t	t	PROPN
ejpam-5587	74	29	(	(	PUNCT
ejpam-5587	74	30	or	or	CCONJ
ejpam-5587	74	31	t	t	PROPN
ejpam-5587	74	32	is	be	AUX
ejpam-5587	74	33	a	a	DET
ejpam-5587	74	34	restriction	restriction	NOUN
ejpam-5587	74	35	of	of	ADP
ejpam-5587	74	36	s	s	NOUN
ejpam-5587	74	37	)	)	PUNCT
ejpam-5587	74	38	,	,	PUNCT
ejpam-5587	74	39	if	if	SCONJ
ejpam-5587	74	40	d(t	d(t	PROPN
ejpam-5587	74	41	)	)	PUNCT
ejpam-5587	74	42	⊆	⊆	NUM
ejpam-5587	74	43	d(s	d(s	PROPN
ejpam-5587	74	44	)	)	PUNCT
ejpam-5587	74	45	and	and	CCONJ
ejpam-5587	74	46	sx	sx	PROPN
ejpam-5587	74	47	=	=	PUNCT
ejpam-5587	74	48	tx	tx	PROPN
ejpam-5587	74	49	for	for	ADP
ejpam-5587	74	50	all	all	DET
ejpam-5587	74	51	x	x	SYM
ejpam-5587	74	52	∈	∈	PROPN
ejpam-5587	74	53	d(t	d(t	PROPN
ejpam-5587	74	54	)	)	PUNCT
ejpam-5587	74	55	.	.	PUNCT
ejpam-5587	75	1	this	this	PRON
ejpam-5587	75	2	is	be	AUX
ejpam-5587	75	3	often	often	ADV
ejpam-5587	75	4	denoted	denote	VERB
ejpam-5587	75	5	as	as	ADP
ejpam-5587	75	6	t	t	PROPN
ejpam-5587	75	7	⊆	⊆	NUM
ejpam-5587	75	8	s.	s.	PROPN
ejpam-5587	75	9	consequently	consequently	ADV
ejpam-5587	75	10	,	,	PUNCT
ejpam-5587	75	11	s	s	NOUN
ejpam-5587	75	12	=	=	X
ejpam-5587	75	13	t	t	X
ejpam-5587	76	1	if	if	SCONJ
ejpam-5587	77	1	and	and	CCONJ
ejpam-5587	77	2	only	only	ADV
ejpam-5587	77	3	if	if	SCONJ
ejpam-5587	77	4	d(s	d(s	PROPN
ejpam-5587	77	5	)	)	PUNCT
ejpam-5587	78	1	=	=	SYM
ejpam-5587	78	2	d(t	d(t	PROPN
ejpam-5587	78	3	)	)	PUNCT
ejpam-5587	78	4	and	and	CCONJ
ejpam-5587	78	5	sx	sx	PROPN
ejpam-5587	78	6	=	=	PUNCT
ejpam-5587	78	7	tx	tx	PROPN
ejpam-5587	78	8	for	for	ADP
ejpam-5587	78	9	all	all	DET
ejpam-5587	78	10	x	x	SYM
ejpam-5587	78	11	∈	∈	PROPN
ejpam-5587	78	12	d(s	d(s	PROPN
ejpam-5587	78	13	)	)	PUNCT
ejpam-5587	79	1	=	=	SYM
ejpam-5587	79	2	d(t	d(t	PROPN
ejpam-5587	79	3	)	)	PUNCT
ejpam-5587	79	4	.	.	PUNCT
ejpam-5587	80	1	definition	definition	NOUN
ejpam-5587	80	2	1	1	NUM
ejpam-5587	80	3	.	.	PUNCT
ejpam-5587	81	1	[	[	X
ejpam-5587	81	2	25	25	NUM
ejpam-5587	81	3	]	]	PUNCT
ejpam-5587	81	4	a	a	DET
ejpam-5587	81	5	densely	densely	ADV
ejpam-5587	81	6	defined	define	VERB
ejpam-5587	81	7	operator	operator	NOUN
ejpam-5587	81	8	t	t	PROPN
ejpam-5587	81	9	∈	∈	PROPN
ejpam-5587	81	10	l(h	l(h	PROPN
ejpam-5587	81	11	)	)	PUNCT
ejpam-5587	81	12	is	be	AUX
ejpam-5587	81	13	said	say	VERB
ejpam-5587	81	14	to	to	PART
ejpam-5587	81	15	be	be	AUX
ejpam-5587	81	16	self	self	NOUN
ejpam-5587	81	17	-	-	PUNCT
ejpam-5587	81	18	adjoint	adjoint	NOUN
ejpam-5587	81	19	operator	operator	NOUN
ejpam-5587	81	20	if	if	SCONJ
ejpam-5587	81	21	d(t	d(t	PROPN
ejpam-5587	81	22	)	)	PUNCT
ejpam-5587	82	1	=	=	PROPN
ejpam-5587	82	2	d(t	d(t	PROPN
ejpam-5587	82	3	∗	∗	NOUN
ejpam-5587	82	4	)	)	PUNCT
ejpam-5587	82	5	and	and	CCONJ
ejpam-5587	82	6	t	t	X
ejpam-5587	82	7	=	=	SYM
ejpam-5587	82	8	t	t	PROPN
ejpam-5587	82	9	∗.	∗.	PROPN
ejpam-5587	82	10	and	and	CCONJ
ejpam-5587	82	11	a	a	DET
ejpam-5587	82	12	self	self	NOUN
ejpam-5587	82	13	-	-	PUNCT
ejpam-5587	82	14	adjoint	adjoint	NOUN
ejpam-5587	82	15	operator	operator	NOUN
ejpam-5587	82	16	t	t	PROPN
ejpam-5587	82	17	is	be	AUX
ejpam-5587	82	18	said	say	VERB
ejpam-5587	82	19	to	to	PART
ejpam-5587	82	20	be	be	AUX
ejpam-5587	82	21	positive	positive	ADJ
ejpam-5587	82	22	if	if	SCONJ
ejpam-5587	82	23	⟨tx	⟨tx	PROPN
ejpam-5587	82	24	,	,	PUNCT
ejpam-5587	82	25	x⟩	x⟩	PUNCT
ejpam-5587	82	26	≥	≥	X
ejpam-5587	82	27	0	0	NUM
ejpam-5587	82	28	for	for	ADP
ejpam-5587	82	29	all	all	DET
ejpam-5587	82	30	x	x	SYM
ejpam-5587	82	31	∈	∈	PROPN
ejpam-5587	82	32	d(t	d(t	PROPN
ejpam-5587	82	33	)	)	PUNCT
ejpam-5587	82	34	.	.	PUNCT
ejpam-5587	83	1	s.	s.	PROPN
ejpam-5587	83	2	alnabulsi	alnabulsi	PROPN
ejpam-5587	83	3	,	,	PUNCT
ejpam-5587	83	4	m.h.m	m.h.m	PROPN
ejpam-5587	83	5	.	.	PUNCT
ejpam-5587	83	6	rashid	rashid	PROPN
ejpam-5587	83	7	/	/	SYM
ejpam-5587	83	8	eur	eur	PROPN
ejpam-5587	83	9	.	.	PUNCT
ejpam-5587	84	1	j.	j.	PROPN
ejpam-5587	84	2	pure	pure	PROPN
ejpam-5587	84	3	appl	appl	PROPN
ejpam-5587	84	4	.	.	PROPN
ejpam-5587	84	5	math	math	PROPN
ejpam-5587	84	6	,	,	PUNCT
ejpam-5587	84	7	18	18	NUM
ejpam-5587	84	8	(	(	PUNCT
ejpam-5587	84	9	1	1	NUM
ejpam-5587	84	10	)	)	PUNCT
ejpam-5587	84	11	(	(	PUNCT
ejpam-5587	84	12	2025	2025	NUM
ejpam-5587	84	13	)	)	PUNCT
ejpam-5587	84	14	,	,	PUNCT
ejpam-5587	84	15	5587	5587	NUM
ejpam-5587	84	16	4	4	NUM
ejpam-5587	84	17	of	of	ADP
ejpam-5587	84	18	20	20	NUM
ejpam-5587	84	19	definition	definition	NOUN
ejpam-5587	84	20	2	2	NUM
ejpam-5587	84	21	.	.	PUNCT
ejpam-5587	85	1	[	[	X
ejpam-5587	85	2	20	20	NUM
ejpam-5587	85	3	,	,	PUNCT
ejpam-5587	85	4	page	page	NOUN
ejpam-5587	85	5	365	365	NUM
ejpam-5587	85	6	]	]	PUNCT
ejpam-5587	85	7	if	if	SCONJ
ejpam-5587	85	8	t	t	PROPN
ejpam-5587	85	9	∈	∈	PROPN
ejpam-5587	85	10	l(h	l(h	PROPN
ejpam-5587	85	11	)	)	PUNCT
ejpam-5587	85	12	is	be	AUX
ejpam-5587	85	13	a	a	DET
ejpam-5587	85	14	closed	closed	ADJ
ejpam-5587	85	15	operator	operator	NOUN
ejpam-5587	85	16	,	,	PUNCT
ejpam-5587	85	17	then	then	ADV
ejpam-5587	85	18	the	the	DET
ejpam-5587	85	19	resolvent	resolvent	ADJ
ejpam-5587	85	20	set	set	NOUN
ejpam-5587	85	21	of	of	ADP
ejpam-5587	85	22	t	t	PROPN
ejpam-5587	85	23	is	be	AUX
ejpam-5587	85	24	defined	define	VERB
ejpam-5587	85	25	by	by	ADP
ejpam-5587	85	26	ρ(t	ρ(t	PROPN
ejpam-5587	85	27	)	)	PUNCT
ejpam-5587	86	1	=	=	PRON
ejpam-5587	86	2	{	{	PUNCT
ejpam-5587	86	3	µ	µ	X
ejpam-5587	86	4	∈	∈	X
ejpam-5587	86	5	c	c	NOUN
ejpam-5587	86	6	:	:	PUNCT
ejpam-5587	87	1	t	t	PROPN
ejpam-5587	87	2	−	−	PROPN
ejpam-5587	87	3	µi	µi	PROPN
ejpam-5587	87	4	is	be	AUX
ejpam-5587	87	5	invertible	invertible	ADJ
ejpam-5587	87	6	and	and	CCONJ
ejpam-5587	87	7	(	(	PUNCT
ejpam-5587	87	8	t	t	NOUN
ejpam-5587	87	9	−	−	NOUN
ejpam-5587	87	10	µi)−1	µi)−1	NOUN
ejpam-5587	87	11	∈	∈	PROPN
ejpam-5587	87	12	b(h	b(h	PROPN
ejpam-5587	87	13	)	)	PUNCT
ejpam-5587	87	14	}	}	PUNCT
ejpam-5587	87	15	and	and	CCONJ
ejpam-5587	87	16	the	the	DET
ejpam-5587	87	17	spectrum	spectrum	NOUN
ejpam-5587	87	18	of	of	ADP
ejpam-5587	87	19	t	t	PROPN
ejpam-5587	87	20	,	,	PUNCT
ejpam-5587	87	21	denoted	denote	VERB
ejpam-5587	87	22	by	by	ADP
ejpam-5587	87	23	σ(t	σ(t	PROPN
ejpam-5587	87	24	)	)	PUNCT
ejpam-5587	87	25	,	,	PUNCT
ejpam-5587	87	26	is	be	AUX
ejpam-5587	87	27	defined	define	VERB
ejpam-5587	87	28	by	by	ADP
ejpam-5587	87	29	σ(t	σ(t	PROPN
ejpam-5587	87	30	)	)	PUNCT
ejpam-5587	87	31	:	:	PUNCT
ejpam-5587	88	1	=	=	PUNCT
ejpam-5587	88	2	c	c	NOUN
ejpam-5587	88	3	\	\	PROPN
ejpam-5587	88	4	ρ(t	ρ(t	PROPN
ejpam-5587	88	5	)	)	PUNCT
ejpam-5587	88	6	note	note	VERB
ejpam-5587	88	7	that	that	SCONJ
ejpam-5587	88	8	σ(t	σ(t	PROPN
ejpam-5587	88	9	)	)	PUNCT
ejpam-5587	88	10	is	be	AUX
ejpam-5587	88	11	a	a	DET
ejpam-5587	88	12	closed	closed	ADJ
ejpam-5587	88	13	subset	subset	NOUN
ejpam-5587	88	14	of	of	ADP
ejpam-5587	88	15	c.	c.	PROPN
ejpam-5587	88	16	moreover	moreover	ADV
ejpam-5587	88	17	σ(t	σ(t	PROPN
ejpam-5587	88	18	)	)	PUNCT
ejpam-5587	88	19	can	can	AUX
ejpam-5587	88	20	be	be	AUX
ejpam-5587	88	21	empty	empty	ADJ
ejpam-5587	88	22	set	set	NOUN
ejpam-5587	88	23	or	or	CCONJ
ejpam-5587	88	24	the	the	DET
ejpam-5587	88	25	whole	whole	ADJ
ejpam-5587	88	26	complex	complex	ADJ
ejpam-5587	88	27	plane	plane	NOUN
ejpam-5587	88	28	c.	c.	NOUN
ejpam-5587	88	29	the	the	DET
ejpam-5587	88	30	spectrum	spectrum	NOUN
ejpam-5587	88	31	of	of	ADP
ejpam-5587	88	32	t	t	PROPN
ejpam-5587	88	33	decomposes	decompose	NOUN
ejpam-5587	88	34	as	as	ADP
ejpam-5587	88	35	the	the	DET
ejpam-5587	88	36	disjoint	disjoint	PROPN
ejpam-5587	88	37	union	union	NOUN
ejpam-5587	88	38	of	of	ADP
ejpam-5587	88	39	the	the	DET
ejpam-5587	88	40	point	point	NOUN
ejpam-5587	88	41	spectrum	spectrum	NOUN
ejpam-5587	88	42	σp(t	σp(t	PUNCT
ejpam-5587	88	43	)	)	PUNCT
ejpam-5587	88	44	,	,	PUNCT
ejpam-5587	88	45	the	the	DET
ejpam-5587	88	46	continuous	continuous	ADJ
ejpam-5587	88	47	spectrum	spectrum	NOUN
ejpam-5587	88	48	σc(t	σc(t	NUM
ejpam-5587	88	49	)	)	PUNCT
ejpam-5587	88	50	and	and	CCONJ
ejpam-5587	88	51	the	the	DET
ejpam-5587	88	52	residual	residual	ADJ
ejpam-5587	88	53	spectrum	spectrum	NOUN
ejpam-5587	88	54	σr(t	σr(t	NOUN
ejpam-5587	88	55	)	)	PUNCT
ejpam-5587	88	56	,	,	PUNCT
ejpam-5587	88	57	where	where	SCONJ
ejpam-5587	88	58	σp(t	σp(t	PUNCT
ejpam-5587	88	59	)	)	PUNCT
ejpam-5587	88	60	=	=	PUNCT
ejpam-5587	88	61	{	{	PUNCT
ejpam-5587	88	62	µ	µ	X
ejpam-5587	88	63	∈	∈	X
ejpam-5587	88	64	c	c	NOUN
ejpam-5587	88	65	:	:	PUNCT
ejpam-5587	89	1	t	t	PROPN
ejpam-5587	89	2	−	−	PROPN
ejpam-5587	89	3	µi	µi	PROPN
ejpam-5587	89	4	is	be	AUX
ejpam-5587	89	5	not	not	PART
ejpam-5587	89	6	injective	injective	ADJ
ejpam-5587	89	7	}	}	PUNCT
ejpam-5587	89	8	,	,	PUNCT
ejpam-5587	89	9	σr(t	σr(t	NOUN
ejpam-5587	89	10	)	)	PUNCT
ejpam-5587	89	11	=	=	PUNCT
ejpam-5587	89	12	{	{	PUNCT
ejpam-5587	89	13	µ	µ	X
ejpam-5587	89	14	∈	∈	X
ejpam-5587	89	15	c	c	NOUN
ejpam-5587	89	16	:	:	PUNCT
ejpam-5587	90	1	t	t	PROPN
ejpam-5587	90	2	−	−	PROPN
ejpam-5587	90	3	µi	µi	PROPN
ejpam-5587	90	4	is	be	AUX
ejpam-5587	90	5	not	not	PART
ejpam-5587	90	6	injective	injective	ADJ
ejpam-5587	90	7	but	but	CCONJ
ejpam-5587	90	8	ran(t	ran(t	NOUN
ejpam-5587	90	9	−	−	PROPN
ejpam-5587	90	10	µi	µi	PROPN
ejpam-5587	90	11	)	)	PUNCT
ejpam-5587	90	12	is	be	AUX
ejpam-5587	90	13	not	not	PART
ejpam-5587	90	14	dense	dense	ADJ
ejpam-5587	90	15	in	in	ADP
ejpam-5587	90	16	h	h	NOUN
ejpam-5587	90	17	}	}	PUNCT
ejpam-5587	90	18	,	,	PUNCT
ejpam-5587	90	19	σc(t	σc(t	X
ejpam-5587	90	20	)	)	PUNCT
ejpam-5587	90	21	=	=	SYM
ejpam-5587	91	1	σ(t	σ(t	PROPN
ejpam-5587	91	2	)	)	PUNCT
ejpam-5587	91	3	\	\	PROPN
ejpam-5587	91	4	(	(	PUNCT
ejpam-5587	91	5	σp(t	σp(t	NUM
ejpam-5587	91	6	)	)	PUNCT
ejpam-5587	91	7	∪	∪	ADP
ejpam-5587	91	8	σr(t	σr(t	NOUN
ejpam-5587	91	9	)	)	PUNCT
ejpam-5587	91	10	)	)	PUNCT
ejpam-5587	91	11	.	.	PUNCT
ejpam-5587	92	1	the	the	DET
ejpam-5587	92	2	spectral	spectral	ADJ
ejpam-5587	92	3	radius	radius	NOUN
ejpam-5587	92	4	of	of	ADP
ejpam-5587	92	5	t	t	PROPN
ejpam-5587	92	6	∈	∈	PROPN
ejpam-5587	92	7	b(h	b(h	PROPN
ejpam-5587	92	8	)	)	PUNCT
ejpam-5587	92	9	is	be	AUX
ejpam-5587	92	10	defined	define	VERB
ejpam-5587	92	11	by	by	ADP
ejpam-5587	92	12	r(t	r(t	NOUN
ejpam-5587	92	13	)	)	PUNCT
ejpam-5587	93	1	:	:	PUNCT
ejpam-5587	93	2	=	=	SYM
ejpam-5587	93	3	sup	sup	INTJ
ejpam-5587	93	4	{	{	PUNCT
ejpam-5587	93	5	|µ|	|µ|	PROPN
ejpam-5587	93	6	:	:	PUNCT
ejpam-5587	93	7	µ	µ	X
ejpam-5587	93	8	∈	∈	NOUN
ejpam-5587	93	9	σ(t	σ(t	PROPN
ejpam-5587	93	10	)	)	PUNCT
ejpam-5587	93	11	}	}	PUNCT
ejpam-5587	93	12	.	.	PUNCT
ejpam-5587	94	1	an	an	DET
ejpam-5587	94	2	operator	operator	NOUN
ejpam-5587	94	3	t	t	PROPN
ejpam-5587	94	4	∈	∈	PROPN
ejpam-5587	94	5	b(h	b(h	PROPN
ejpam-5587	94	6	)	)	PUNCT
ejpam-5587	94	7	is	be	AUX
ejpam-5587	94	8	said	say	VERB
ejpam-5587	94	9	to	to	PART
ejpam-5587	94	10	be	be	AUX
ejpam-5587	94	11	normaloid	normaloid	NOUN
ejpam-5587	94	12	,	,	PUNCT
ejpam-5587	94	13	if	if	SCONJ
ejpam-5587	94	14	r(t	r(t	NOUN
ejpam-5587	94	15	)	)	PUNCT
ejpam-5587	95	1	=	=	PRON
ejpam-5587	95	2	∥t∥.	∥t∥.	PROPN
ejpam-5587	95	3	recall	recall	VERB
ejpam-5587	95	4	that	that	SCONJ
ejpam-5587	95	5	a	a	DET
ejpam-5587	95	6	linear	linear	ADJ
ejpam-5587	95	7	operator	operator	NOUN
ejpam-5587	95	8	t	t	PROPN
ejpam-5587	95	9	∈	∈	PROPN
ejpam-5587	95	10	l(h	l(h	PROPN
ejpam-5587	95	11	)	)	PUNCT
ejpam-5587	95	12	is	be	AUX
ejpam-5587	95	13	compact	compact	ADJ
ejpam-5587	95	14	,	,	PUNCT
ejpam-5587	95	15	if	if	SCONJ
ejpam-5587	95	16	t	t	PROPN
ejpam-5587	95	17	maps	map	VERB
ejpam-5587	95	18	every	every	DET
ejpam-5587	95	19	bounded	bound	VERB
ejpam-5587	95	20	set	set	VERB
ejpam-5587	95	21	in	in	ADP
ejpam-5587	95	22	h	h	NOUN
ejpam-5587	95	23	to	to	ADP
ejpam-5587	95	24	a	a	DET
ejpam-5587	95	25	pre	pre	ADJ
ejpam-5587	95	26	-	-	ADJ
ejpam-5587	95	27	compact	compact	ADJ
ejpam-5587	95	28	set	set	NOUN
ejpam-5587	95	29	in	in	ADP
ejpam-5587	95	30	h.	h.	NOUN
ejpam-5587	95	31	for	for	ADP
ejpam-5587	95	32	more	more	ADJ
ejpam-5587	95	33	details	detail	NOUN
ejpam-5587	95	34	about	about	ADP
ejpam-5587	95	35	compact	compact	ADJ
ejpam-5587	95	36	operators	operator	NOUN
ejpam-5587	95	37	,	,	PUNCT
ejpam-5587	95	38	we	we	PRON
ejpam-5587	95	39	refer	refer	VERB
ejpam-5587	95	40	to	to	ADP
ejpam-5587	95	41	[	[	X
ejpam-5587	95	42	21	21	NUM
ejpam-5587	95	43	]	]	PUNCT
ejpam-5587	95	44	.	.	PUNCT
ejpam-5587	96	1	definition	definition	NOUN
ejpam-5587	96	2	3	3	NUM
ejpam-5587	96	3	.	.	PUNCT
ejpam-5587	97	1	[	[	X
ejpam-5587	97	2	21	21	NUM
ejpam-5587	97	3	,	,	PUNCT
ejpam-5587	97	4	page	page	NOUN
ejpam-5587	97	5	156	156	NUM
ejpam-5587	97	6	]	]	PUNCT
ejpam-5587	97	7	a	a	DET
ejpam-5587	97	8	closed	closed	ADJ
ejpam-5587	97	9	operator	operator	NOUN
ejpam-5587	97	10	t	t	NOUN
ejpam-5587	97	11	in	in	ADP
ejpam-5587	97	12	a	a	DET
ejpam-5587	97	13	densely	densely	ADV
ejpam-5587	97	14	defined	define	VERB
ejpam-5587	97	15	space	space	NOUN
ejpam-5587	97	16	h	h	NOUN
ejpam-5587	97	17	is	be	AUX
ejpam-5587	97	18	termed	term	VERB
ejpam-5587	97	19	fredholm	fredholm	NOUN
ejpam-5587	97	20	if	if	SCONJ
ejpam-5587	97	21	ran(t	ran(t	PROPN
ejpam-5587	97	22	)	)	PUNCT
ejpam-5587	97	23	is	be	AUX
ejpam-5587	97	24	closed	close	VERB
ejpam-5587	97	25	,	,	PUNCT
ejpam-5587	97	26	and	and	CCONJ
ejpam-5587	97	27	both	both	CCONJ
ejpam-5587	97	28	the	the	DET
ejpam-5587	97	29	dimensions	dimension	NOUN
ejpam-5587	97	30	of	of	ADP
ejpam-5587	97	31	ker(t	ker(t	NOUN
ejpam-5587	97	32	)	)	PUNCT
ejpam-5587	97	33	and	and	CCONJ
ejpam-5587	97	34	its	its	PRON
ejpam-5587	97	35	orthogonal	orthogonal	ADJ
ejpam-5587	97	36	complement	complement	NOUN
ejpam-5587	97	37	ran(t	ran(t	NOUN
ejpam-5587	97	38	)	)	PUNCT
ejpam-5587	98	1	⊥	⊥	NOUN
ejpam-5587	98	2	are	be	AUX
ejpam-5587	98	3	finite	finite	ADJ
ejpam-5587	98	4	.	.	PUNCT
ejpam-5587	99	1	in	in	ADP
ejpam-5587	99	2	such	such	ADJ
ejpam-5587	99	3	instances	instance	NOUN
ejpam-5587	99	4	,	,	PUNCT
ejpam-5587	99	5	the	the	DET
ejpam-5587	99	6	index	index	NOUN
ejpam-5587	99	7	of	of	ADP
ejpam-5587	99	8	t	t	PROPN
ejpam-5587	99	9	,	,	PUNCT
ejpam-5587	99	10	denoted	denote	VERB
ejpam-5587	99	11	by	by	ADP
ejpam-5587	99	12	ind(t	ind(t	PROPN
ejpam-5587	99	13	)	)	PUNCT
ejpam-5587	99	14	,	,	PUNCT
ejpam-5587	99	15	is	be	AUX
ejpam-5587	99	16	defined	define	VERB
ejpam-5587	99	17	as	as	ADP
ejpam-5587	99	18	ind(t	ind(t	ADJ
ejpam-5587	99	19	)	)	PUNCT
ejpam-5587	99	20	=	=	SYM
ejpam-5587	99	21	dim(ker(t	dim(ker(t	NOUN
ejpam-5587	99	22	)	)	PUNCT
ejpam-5587	99	23	)	)	PUNCT
ejpam-5587	100	1	−	−	PROPN
ejpam-5587	100	2	dim(ran(t	dim(ran(t	SYM
ejpam-5587	100	3	)	)	PUNCT
ejpam-5587	100	4	⊥	⊥	NUM
ejpam-5587	100	5	)	)	PUNCT
ejpam-5587	100	6	.	.	PUNCT
ejpam-5587	101	1	remark	remark	NOUN
ejpam-5587	101	2	2	2	NUM
ejpam-5587	101	3	.	.	PUNCT
ejpam-5587	102	1	if	if	SCONJ
ejpam-5587	102	2	t	t	PROPN
ejpam-5587	102	3	∈	∈	PROPN
ejpam-5587	102	4	l(h	l(h	PROPN
ejpam-5587	102	5	)	)	PUNCT
ejpam-5587	102	6	is	be	AUX
ejpam-5587	102	7	a	a	DET
ejpam-5587	102	8	densely	densely	ADV
ejpam-5587	102	9	defined	define	VERB
ejpam-5587	102	10	closed	closed	ADJ
ejpam-5587	102	11	fredholm	fredholm	NOUN
ejpam-5587	102	12	operator	operator	NOUN
ejpam-5587	102	13	and	and	CCONJ
ejpam-5587	102	14	k	k	PROPN
ejpam-5587	102	15	is	be	AUX
ejpam-5587	102	16	a	a	DET
ejpam-5587	102	17	compact	compact	ADJ
ejpam-5587	102	18	operator	operator	NOUN
ejpam-5587	102	19	,	,	PUNCT
ejpam-5587	102	20	then	then	ADV
ejpam-5587	102	21	t	t	PROPN
ejpam-5587	103	1	+	+	PROPN
ejpam-5587	103	2	k	k	PROPN
ejpam-5587	103	3	is	be	AUX
ejpam-5587	103	4	also	also	ADV
ejpam-5587	103	5	fredholm	fredholm	NOUN
ejpam-5587	103	6	and	and	CCONJ
ejpam-5587	103	7	ind(t	ind(t	ADJ
ejpam-5587	103	8	+	+	PROPN
ejpam-5587	103	9	k	k	NOUN
ejpam-5587	103	10	)	)	PUNCT
ejpam-5587	103	11	=	=	SYM
ejpam-5587	103	12	ind(t	ind(t	ADJ
ejpam-5587	103	13	)	)	PUNCT
ejpam-5587	103	14	.	.	PUNCT
ejpam-5587	104	1	definition	definition	NOUN
ejpam-5587	104	2	4	4	NUM
ejpam-5587	104	3	.	.	PUNCT
ejpam-5587	105	1	[	[	X
ejpam-5587	105	2	21	21	NUM
ejpam-5587	105	3	,	,	PUNCT
ejpam-5587	105	4	page	page	NOUN
ejpam-5587	105	5	172	172	NUM
ejpam-5587	105	6	]	]	PUNCT
ejpam-5587	105	7	if	if	SCONJ
ejpam-5587	105	8	t	t	PROPN
ejpam-5587	105	9	∈	∈	PROPN
ejpam-5587	105	10	l(h	l(h	PROPN
ejpam-5587	105	11	)	)	PUNCT
ejpam-5587	105	12	is	be	AUX
ejpam-5587	105	13	a	a	DET
ejpam-5587	105	14	densely	densely	ADV
ejpam-5587	105	15	defined	define	VERB
ejpam-5587	105	16	closed	closed	ADJ
ejpam-5587	105	17	operator	operator	NOUN
ejpam-5587	105	18	,	,	PUNCT
ejpam-5587	105	19	then	then	ADV
ejpam-5587	105	20	the	the	DET
ejpam-5587	105	21	weyl	weyl	PROPN
ejpam-5587	105	22	’s	’s	PART
ejpam-5587	105	23	spectrum	spectrum	NOUN
ejpam-5587	105	24	of	of	ADP
ejpam-5587	105	25	t	t	PROPN
ejpam-5587	105	26	is	be	AUX
ejpam-5587	105	27	defined	define	VERB
ejpam-5587	105	28	by	by	ADP
ejpam-5587	105	29	σw(t	σw(t	PUNCT
ejpam-5587	105	30	)	)	PUNCT
ejpam-5587	105	31	=	=	SYM
ejpam-5587	106	1	{	{	PUNCT
ejpam-5587	106	2	λ	λ	X
ejpam-5587	106	3	∈	∈	NOUN
ejpam-5587	106	4	c	c	NOUN
ejpam-5587	106	5	:	:	PUNCT
ejpam-5587	106	6	t	t	PROPN
ejpam-5587	106	7	−	−	PROPN
ejpam-5587	107	1	λi	λi	INTJ
ejpam-5587	107	2	is	be	AUX
ejpam-5587	107	3	not	not	PART
ejpam-5587	107	4	fredholm	fredholm	NOUN
ejpam-5587	107	5	of	of	ADP
ejpam-5587	107	6	index	index	NOUN
ejpam-5587	107	7	0	0	NUM
ejpam-5587	107	8	}	}	PUNCT
ejpam-5587	107	9	and	and	CCONJ
ejpam-5587	107	10	π00(t	π00(t	VERB
ejpam-5587	107	11	)	)	PUNCT
ejpam-5587	108	1	=	=	SYM
ejpam-5587	108	2	{	{	PUNCT
ejpam-5587	108	3	λ	λ	X
ejpam-5587	108	4	∈	∈	PROPN
ejpam-5587	108	5	σp(t	σp(t	PUNCT
ejpam-5587	108	6	)	)	PUNCT
ejpam-5587	108	7	:	:	PUNCT
ejpam-5587	109	1	λ	λ	NOUN
ejpam-5587	109	2	is	be	AUX
ejpam-5587	109	3	isolated	isolate	VERB
ejpam-5587	109	4	with	with	ADP
ejpam-5587	109	5	dim	dim	ADJ
ejpam-5587	109	6	(	(	PUNCT
ejpam-5587	109	7	ker(t	ker(t	NOUN
ejpam-5587	109	8	−	−	NUM
ejpam-5587	109	9	λi	λi	NOUN
ejpam-5587	109	10	)	)	PUNCT
ejpam-5587	109	11	)	)	PUNCT
ejpam-5587	110	1	<	<	X
ejpam-5587	111	1	+	+	PUNCT
ejpam-5587	111	2	∞	∞	NOUN
ejpam-5587	111	3	}	}	PUNCT
ejpam-5587	111	4	.	.	PUNCT
ejpam-5587	112	1	suppose	suppose	VERB
ejpam-5587	112	2	t	t	PROPN
ejpam-5587	112	3	∈	∈	PROPN
ejpam-5587	112	4	l(h	l(h	PROPN
ejpam-5587	112	5	)	)	PUNCT
ejpam-5587	112	6	is	be	AUX
ejpam-5587	112	7	a	a	DET
ejpam-5587	112	8	densely	densely	ADV
ejpam-5587	112	9	defined	define	VERB
ejpam-5587	112	10	closed	closed	ADJ
ejpam-5587	112	11	operator	operator	NOUN
ejpam-5587	112	12	with	with	ADP
ejpam-5587	112	13	σ(t	σ(t	NOUN
ejpam-5587	112	14	)	)	PUNCT
ejpam-5587	113	1	=	=	PUNCT
ejpam-5587	113	2	σ	σ	PROPN
ejpam-5587	113	3	∪	∪	X
ejpam-5587	113	4	τ	τ	PROPN
ejpam-5587	113	5	,	,	PUNCT
ejpam-5587	113	6	where	where	SCONJ
ejpam-5587	113	7	σ	σ	PROPN
ejpam-5587	113	8	is	be	AUX
ejpam-5587	113	9	contained	contain	VERB
ejpam-5587	113	10	in	in	ADP
ejpam-5587	113	11	some	some	DET
ejpam-5587	113	12	bounded	bounded	ADJ
ejpam-5587	113	13	domain	domain	NOUN
ejpam-5587	113	14	∆	∆	PROPN
ejpam-5587	113	15	and	and	CCONJ
ejpam-5587	113	16	τ	τ	PROPN
ejpam-5587	113	17	is	be	AUX
ejpam-5587	113	18	a	a	DET
ejpam-5587	113	19	subset	subset	NOUN
ejpam-5587	113	20	of	of	ADP
ejpam-5587	113	21	the	the	DET
ejpam-5587	113	22	complement	complement	NOUN
ejpam-5587	113	23	of	of	ADP
ejpam-5587	113	24	∆.	∆.	NOUN
ejpam-5587	113	25	let	let	VERB
ejpam-5587	113	26	λ	λ	NOUN
ejpam-5587	113	27	be	be	AUX
ejpam-5587	113	28	the	the	DET
ejpam-5587	113	29	boundary	boundary	NOUN
ejpam-5587	113	30	of	of	ADP
ejpam-5587	113	31	∆	∆	PROPN
ejpam-5587	113	32	,	,	PUNCT
ejpam-5587	113	33	then	then	ADV
ejpam-5587	113	34	eσ	eσ	ADP
ejpam-5587	113	35	=	=	SYM
ejpam-5587	113	36	1	1	NUM
ejpam-5587	113	37	2πi	2πi	NOUN
ejpam-5587	113	38	∫	∫	PROPN
ejpam-5587	113	39	λ	λ	PROPN
ejpam-5587	113	40	(	(	PUNCT
ejpam-5587	113	41	zi	zi	NOUN
ejpam-5587	113	42	−	−	PROPN
ejpam-5587	113	43	t	t	NOUN
ejpam-5587	113	44	)	)	PUNCT
ejpam-5587	113	45	−1dz	−1dz	NOUN
ejpam-5587	113	46	(	(	PUNCT
ejpam-5587	113	47	1	1	X
ejpam-5587	113	48	)	)	PUNCT
ejpam-5587	113	49	is	be	AUX
ejpam-5587	113	50	called	call	VERB
ejpam-5587	113	51	the	the	DET
ejpam-5587	113	52	riesz	riesz	NOUN
ejpam-5587	113	53	projection	projection	NOUN
ejpam-5587	113	54	with	with	ADP
ejpam-5587	113	55	respect	respect	NOUN
ejpam-5587	113	56	to	to	ADP
ejpam-5587	113	57	σ	σ	PROPN
ejpam-5587	113	58	.	.	PUNCT
ejpam-5587	114	1	s.	s.	PROPN
ejpam-5587	114	2	alnabulsi	alnabulsi	PROPN
ejpam-5587	114	3	,	,	PUNCT
ejpam-5587	114	4	m.h.m	m.h.m	PROPN
ejpam-5587	114	5	.	.	PUNCT
ejpam-5587	114	6	rashid	rashid	PROPN
ejpam-5587	114	7	/	/	SYM
ejpam-5587	114	8	eur	eur	PROPN
ejpam-5587	114	9	.	.	PUNCT
ejpam-5587	115	1	j.	j.	PROPN
ejpam-5587	115	2	pure	pure	PROPN
ejpam-5587	115	3	appl	appl	PROPN
ejpam-5587	115	4	.	.	PROPN
ejpam-5587	115	5	math	math	PROPN
ejpam-5587	115	6	,	,	PUNCT
ejpam-5587	115	7	18	18	NUM
ejpam-5587	115	8	(	(	PUNCT
ejpam-5587	115	9	1	1	NUM
ejpam-5587	115	10	)	)	PUNCT
ejpam-5587	115	11	(	(	PUNCT
ejpam-5587	115	12	2025	2025	NUM
ejpam-5587	115	13	)	)	PUNCT
ejpam-5587	115	14	,	,	PUNCT
ejpam-5587	115	15	5587	5587	NUM
ejpam-5587	115	16	5	5	NUM
ejpam-5587	115	17	of	of	ADP
ejpam-5587	115	18	20	20	NUM
ejpam-5587	115	19	theorem	theorem	NOUN
ejpam-5587	115	20	1	1	NUM
ejpam-5587	115	21	.	.	PUNCT
ejpam-5587	116	1	[	[	X
ejpam-5587	116	2	12	12	NUM
ejpam-5587	116	3	,	,	PUNCT
ejpam-5587	116	4	theorem	theorem	VERB
ejpam-5587	116	5	2.1	2.1	NUM
ejpam-5587	116	6	,	,	PUNCT
ejpam-5587	116	7	page	page	NOUN
ejpam-5587	116	8	326	326	NUM
ejpam-5587	116	9	]	]	PUNCT
ejpam-5587	116	10	suppose	suppose	VERB
ejpam-5587	116	11	t	t	PROPN
ejpam-5587	116	12	∈	∈	PROPN
ejpam-5587	116	13	l(h	l(h	PROPN
ejpam-5587	116	14	)	)	PUNCT
ejpam-5587	116	15	is	be	AUX
ejpam-5587	116	16	a	a	DET
ejpam-5587	116	17	densely	densely	ADV
ejpam-5587	116	18	defined	define	VERB
ejpam-5587	116	19	closed	closed	ADJ
ejpam-5587	116	20	operator	operator	NOUN
ejpam-5587	116	21	with	with	ADP
ejpam-5587	116	22	σ(t	σ(t	NOUN
ejpam-5587	116	23	)	)	PUNCT
ejpam-5587	117	1	=	=	PUNCT
ejpam-5587	117	2	σ	σ	PROPN
ejpam-5587	117	3	∪	∪	X
ejpam-5587	117	4	τ	τ	PROPN
ejpam-5587	117	5	,	,	PUNCT
ejpam-5587	117	6	where	where	SCONJ
ejpam-5587	117	7	σ	σ	PROPN
ejpam-5587	117	8	is	be	AUX
ejpam-5587	117	9	contained	contain	VERB
ejpam-5587	117	10	in	in	ADP
ejpam-5587	117	11	some	some	DET
ejpam-5587	117	12	bounded	bounded	ADJ
ejpam-5587	117	13	domain	domain	NOUN
ejpam-5587	117	14	and	and	CCONJ
ejpam-5587	117	15	eσ	eσ	NOUN
ejpam-5587	117	16	is	be	AUX
ejpam-5587	117	17	the	the	DET
ejpam-5587	117	18	operator	operator	NOUN
ejpam-5587	117	19	defined	define	VERB
ejpam-5587	117	20	in	in	ADP
ejpam-5587	117	21	equation	equation	NOUN
ejpam-5587	117	22	(	(	PUNCT
ejpam-5587	117	23	1	1	NUM
ejpam-5587	117	24	)	)	PUNCT
ejpam-5587	117	25	.	.	PUNCT
ejpam-5587	118	1	then	then	ADV
ejpam-5587	118	2	(	(	PUNCT
ejpam-5587	118	3	i	i	NOUN
ejpam-5587	118	4	)	)	PUNCT
ejpam-5587	118	5	eσ	eσ	ADP
ejpam-5587	118	6	is	be	AUX
ejpam-5587	118	7	a	a	DET
ejpam-5587	118	8	projection	projection	NOUN
ejpam-5587	118	9	.	.	PUNCT
ejpam-5587	119	1	(	(	PUNCT
ejpam-5587	119	2	ii	ii	NOUN
ejpam-5587	119	3	)	)	PUNCT
ejpam-5587	119	4	the	the	DET
ejpam-5587	119	5	subspaces	subspace	NOUN
ejpam-5587	119	6	ran(eσ	ran(eσ	NUM
ejpam-5587	119	7	)	)	PUNCT
ejpam-5587	119	8	and	and	CCONJ
ejpam-5587	119	9	ker(eσ	ker(eσ	NOUN
ejpam-5587	119	10	)	)	PUNCT
ejpam-5587	119	11	are	be	AUX
ejpam-5587	119	12	invariant	invariant	ADJ
ejpam-5587	119	13	under	under	ADP
ejpam-5587	119	14	t	t	PROPN
ejpam-5587	119	15	.	.	PUNCT
ejpam-5587	120	1	(	(	PUNCT
ejpam-5587	120	2	iii	iii	X
ejpam-5587	120	3	)	)	PUNCT
ejpam-5587	120	4	the	the	DET
ejpam-5587	120	5	subspace	subspace	PROPN
ejpam-5587	120	6	ran(eσ	ran(eσ	NOUN
ejpam-5587	120	7	)	)	PUNCT
ejpam-5587	120	8	is	be	AUX
ejpam-5587	120	9	contained	contain	VERB
ejpam-5587	120	10	in	in	ADP
ejpam-5587	120	11	d(t	d(t	PROPN
ejpam-5587	120	12	)	)	PUNCT
ejpam-5587	120	13	and	and	CCONJ
ejpam-5587	120	14	t	t	PROPN
ejpam-5587	120	15	|ran(eσ	|ran(eσ	PROPN
ejpam-5587	120	16	)	)	PUNCT
ejpam-5587	120	17	is	be	AUX
ejpam-5587	120	18	bounded	bound	VERB
ejpam-5587	120	19	.	.	PUNCT
ejpam-5587	121	1	(	(	PUNCT
ejpam-5587	121	2	iv	iv	X
ejpam-5587	121	3	)	)	PUNCT
ejpam-5587	121	4	σ	σ	PROPN
ejpam-5587	121	5	(	(	PUNCT
ejpam-5587	121	6	t	t	NOUN
ejpam-5587	121	7	|ran(eσ	|ran(eσ	NUM
ejpam-5587	121	8	)	)	PUNCT
ejpam-5587	121	9	)	)	PUNCT
ejpam-5587	122	1	=	=	SYM
ejpam-5587	122	2	σ	σ	PROPN
ejpam-5587	122	3	and	and	CCONJ
ejpam-5587	122	4	σ(t	σ(t	PROPN
ejpam-5587	122	5	|ker(eσ	|ker(eσ	NUM
ejpam-5587	122	6	)	)	PUNCT
ejpam-5587	122	7	)	)	PUNCT
ejpam-5587	123	1	=	=	SYM
ejpam-5587	123	2	τ	τ	PROPN
ejpam-5587	123	3	.	.	PUNCT
ejpam-5587	124	1	in	in	ADP
ejpam-5587	124	2	particular	particular	ADJ
ejpam-5587	124	3	,	,	PUNCT
ejpam-5587	124	4	if	if	SCONJ
ejpam-5587	124	5	µ	µ	NOUN
ejpam-5587	124	6	is	be	AUX
ejpam-5587	124	7	an	an	DET
ejpam-5587	124	8	isolated	isolated	ADJ
ejpam-5587	124	9	point	point	NOUN
ejpam-5587	124	10	of	of	ADP
ejpam-5587	124	11	σ(t	σ(t	PROPN
ejpam-5587	124	12	)	)	PUNCT
ejpam-5587	124	13	,	,	PUNCT
ejpam-5587	124	14	then	then	ADV
ejpam-5587	124	15	there	there	PRON
ejpam-5587	124	16	exist	exist	VERB
ejpam-5587	124	17	a	a	DET
ejpam-5587	124	18	positive	positive	ADJ
ejpam-5587	124	19	real	real	ADJ
ejpam-5587	124	20	number	number	NOUN
ejpam-5587	124	21	r	r	NOUN
ejpam-5587	124	22	such	such	ADJ
ejpam-5587	124	23	that	that	SCONJ
ejpam-5587	124	24	{	{	PUNCT
ejpam-5587	124	25	z	z	NOUN
ejpam-5587	124	26	∈	∈	PROPN
ejpam-5587	124	27	c	c	NOUN
ejpam-5587	124	28	:	:	PUNCT
ejpam-5587	124	29	|z−µ|	|z−µ|	NUM
ejpam-5587	124	30	≤	≤	NUM
ejpam-5587	124	31	r	r	NOUN
ejpam-5587	124	32	}	}	PUNCT
ejpam-5587	124	33	∩	∩	NOUN
ejpam-5587	124	34	σ(t	σ(t	X
ejpam-5587	124	35	)	)	PUNCT
ejpam-5587	125	1	=	=	PUNCT
ejpam-5587	125	2	{	{	PUNCT
ejpam-5587	125	3	µ	µ	NOUN
ejpam-5587	125	4	}	}	PUNCT
ejpam-5587	125	5	.	.	PUNCT
ejpam-5587	126	1	if	if	SCONJ
ejpam-5587	126	2	we	we	PRON
ejpam-5587	126	3	take	take	VERB
ejpam-5587	126	4	λ	λ	X
ejpam-5587	126	5	=	=	PUNCT
ejpam-5587	126	6	{	{	PUNCT
ejpam-5587	126	7	z	z	NOUN
ejpam-5587	126	8	∈	∈	PROPN
ejpam-5587	126	9	c	c	NOUN
ejpam-5587	126	10	:	:	PUNCT
ejpam-5587	127	1	|z	|z	PROPN
ejpam-5587	128	1	−	−	PROPN
ejpam-5587	128	2	µ|	µ|	PROPN
ejpam-5587	128	3	=	=	SYM
ejpam-5587	128	4	r	r	X
ejpam-5587	128	5	}	}	PUNCT
ejpam-5587	128	6	,	,	PUNCT
ejpam-5587	128	7	then	then	ADV
ejpam-5587	128	8	the	the	DET
ejpam-5587	128	9	riesz	riesz	PROPN
ejpam-5587	128	10	projection	projection	NOUN
ejpam-5587	128	11	with	with	ADP
ejpam-5587	128	12	respect	respect	NOUN
ejpam-5587	128	13	to	to	ADP
ejpam-5587	128	14	µ	µ	NOUN
ejpam-5587	128	15	is	be	AUX
ejpam-5587	128	16	defined	define	VERB
ejpam-5587	128	17	by	by	ADP
ejpam-5587	128	18	eµ	eµ	NOUN
ejpam-5587	128	19	=	=	SYM
ejpam-5587	129	1	1	1	NUM
ejpam-5587	129	2	2πi	2πi	ADJ
ejpam-5587	129	3	∫	∫	PROPN
ejpam-5587	129	4	λ	λ	PROPN
ejpam-5587	129	5	(	(	PUNCT
ejpam-5587	129	6	zi	zi	NOUN
ejpam-5587	129	7	−	−	PROPN
ejpam-5587	129	8	t	t	PROPN
ejpam-5587	129	9	)	)	PUNCT
ejpam-5587	129	10	−1dz	−1dz	NOUN
ejpam-5587	129	11	.	.	PUNCT
ejpam-5587	130	1	(	(	PUNCT
ejpam-5587	130	2	2	2	X
ejpam-5587	130	3	)	)	PUNCT
ejpam-5587	130	4	definition	definition	NOUN
ejpam-5587	130	5	5	5	NUM
ejpam-5587	130	6	.	.	PUNCT
ejpam-5587	131	1	[	[	X
ejpam-5587	131	2	16	16	NUM
ejpam-5587	131	3	]	]	PUNCT
ejpam-5587	131	4	let	let	VERB
ejpam-5587	131	5	t	t	PROPN
ejpam-5587	131	6	∈	∈	PROPN
ejpam-5587	131	7	l(h	l(h	PROPN
ejpam-5587	131	8	)	)	PUNCT
ejpam-5587	131	9	be	be	AUX
ejpam-5587	131	10	a	a	DET
ejpam-5587	131	11	closed	closed	ADJ
ejpam-5587	131	12	operator	operator	NOUN
ejpam-5587	131	13	.	.	PUNCT
ejpam-5587	132	1	then	then	ADV
ejpam-5587	132	2	(	(	PUNCT
ejpam-5587	132	3	i	i	NOUN
ejpam-5587	132	4	)	)	PUNCT
ejpam-5587	132	5	the	the	DET
ejpam-5587	132	6	minimum	minimum	ADJ
ejpam-5587	132	7	modulus	modulus	NOUN
ejpam-5587	132	8	of	of	ADP
ejpam-5587	132	9	t	t	PROPN
ejpam-5587	132	10	is	be	AUX
ejpam-5587	132	11	defined	define	VERB
ejpam-5587	132	12	by	by	ADP
ejpam-5587	132	13	m(t	m(t	PROPN
ejpam-5587	132	14	)	)	PUNCT
ejpam-5587	133	1	:	:	PUNCT
ejpam-5587	133	2	=	=	SYM
ejpam-5587	133	3	inf	inf	PROPN
ejpam-5587	133	4	{	{	PUNCT
ejpam-5587	133	5	∥tx∥	∥tx∥	NOUN
ejpam-5587	133	6	:	:	PUNCT
ejpam-5587	133	7	x	x	X
ejpam-5587	133	8	∈	∈	PROPN
ejpam-5587	133	9	td(t	td(t	NOUN
ejpam-5587	133	10	)	)	PUNCT
ejpam-5587	133	11	}	}	PUNCT
ejpam-5587	133	12	.	.	PUNCT
ejpam-5587	134	1	then	then	ADV
ejpam-5587	134	2	(	(	PUNCT
ejpam-5587	134	3	ii	ii	NOUN
ejpam-5587	134	4	)	)	PUNCT
ejpam-5587	134	5	the	the	DET
ejpam-5587	134	6	reduced	reduce	VERB
ejpam-5587	134	7	minimum	minimum	ADJ
ejpam-5587	134	8	modulus	modulus	NOUN
ejpam-5587	134	9	of	of	ADP
ejpam-5587	134	10	t	t	NOUN
ejpam-5587	134	11	isdenoted	isdenote	VERB
ejpam-5587	134	12	by	by	ADP
ejpam-5587	134	13	γ(t	γ(t	NOUN
ejpam-5587	134	14	)	)	PUNCT
ejpam-5587	134	15	:	:	PUNCT
ejpam-5587	134	16	=	=	SYM
ejpam-5587	134	17	inf	inf	PROPN
ejpam-5587	134	18	{	{	PUNCT
ejpam-5587	134	19	∥tx∥	∥tx∥	NOUN
ejpam-5587	134	20	:	:	PUNCT
ejpam-5587	134	21	x	x	X
ejpam-5587	134	22	∈	∈	NOUN
ejpam-5587	134	23	td(t	td(t	NUM
ejpam-5587	134	24	)	)	PUNCT
ejpam-5587	134	25	∩ker(t	∩ker(t	NUM
ejpam-5587	134	26	)	)	PUNCT
ejpam-5587	134	27	⊥	⊥	NOUN
ejpam-5587	134	28	}	}	PUNCT
ejpam-5587	134	29	.	.	PUNCT
ejpam-5587	135	1	by	by	ADP
ejpam-5587	135	2	the	the	DET
ejpam-5587	135	3	definition	definition	NOUN
ejpam-5587	135	4	,	,	PUNCT
ejpam-5587	135	5	it	it	PRON
ejpam-5587	135	6	is	be	AUX
ejpam-5587	135	7	clear	clear	ADJ
ejpam-5587	135	8	that	that	SCONJ
ejpam-5587	135	9	m(t	m(t	NOUN
ejpam-5587	135	10	)	)	PUNCT
ejpam-5587	135	11	≤	≤	NUM
ejpam-5587	136	1	γ(t	γ(t	VERB
ejpam-5587	136	2	)	)	PUNCT
ejpam-5587	137	1	the	the	DET
ejpam-5587	137	2	following	follow	VERB
ejpam-5587	137	3	characterization	characterization	NOUN
ejpam-5587	137	4	of	of	ADP
ejpam-5587	137	5	closed	closed	ADJ
ejpam-5587	137	6	range	range	NOUN
ejpam-5587	137	7	operators	operator	NOUN
ejpam-5587	137	8	is	be	AUX
ejpam-5587	137	9	frequently	frequently	ADV
ejpam-5587	137	10	used	use	VERB
ejpam-5587	137	11	in	in	ADP
ejpam-5587	137	12	the	the	DET
ejpam-5587	137	13	article	article	NOUN
ejpam-5587	137	14	.	.	PUNCT
ejpam-5587	138	1	theorem	theorem	PROPN
ejpam-5587	138	2	2	2	NUM
ejpam-5587	138	3	.	.	PUNCT
ejpam-5587	139	1	[	[	X
ejpam-5587	139	2	1	1	NUM
ejpam-5587	139	3	,	,	PUNCT
ejpam-5587	139	4	page	page	NOUN
ejpam-5587	139	5	334	334	NUM
ejpam-5587	139	6	]	]	PUNCT
ejpam-5587	139	7	for	for	ADP
ejpam-5587	139	8	a	a	DET
ejpam-5587	139	9	densely	densely	ADV
ejpam-5587	139	10	defined	define	VERB
ejpam-5587	139	11	closed	closed	ADJ
ejpam-5587	139	12	operator	operator	NOUN
ejpam-5587	139	13	t	t	PROPN
ejpam-5587	139	14	∈	∈	PROPN
ejpam-5587	139	15	l(h	l(h	PROPN
ejpam-5587	139	16	)	)	PUNCT
ejpam-5587	139	17	,	,	PUNCT
ejpam-5587	139	18	the	the	DET
ejpam-5587	139	19	following	follow	VERB
ejpam-5587	139	20	are	be	AUX
ejpam-5587	139	21	equivalent	equivalent	ADJ
ejpam-5587	139	22	.	.	PUNCT
ejpam-5587	140	1	(	(	PUNCT
ejpam-5587	140	2	i	i	NOUN
ejpam-5587	140	3	)	)	PUNCT
ejpam-5587	140	4	ran(t	ran(t	PROPN
ejpam-5587	140	5	)	)	PUNCT
ejpam-5587	140	6	is	be	AUX
ejpam-5587	140	7	closed	closed	ADJ
ejpam-5587	140	8	.	.	PUNCT
ejpam-5587	141	1	(	(	PUNCT
ejpam-5587	141	2	ii	ii	NOUN
ejpam-5587	141	3	)	)	PUNCT
ejpam-5587	141	4	ran(t	ran(t	NOUN
ejpam-5587	141	5	∗	∗	NOUN
ejpam-5587	141	6	)	)	PUNCT
ejpam-5587	141	7	is	be	AUX
ejpam-5587	141	8	closed	closed	ADJ
ejpam-5587	141	9	.	.	PUNCT
ejpam-5587	142	1	(	(	PUNCT
ejpam-5587	142	2	iii	iii	NOUN
ejpam-5587	142	3	)	)	PUNCT
ejpam-5587	142	4	γ(t	γ(t	NOUN
ejpam-5587	142	5	)	)	PUNCT
ejpam-5587	142	6	>	>	X
ejpam-5587	142	7	0	0	X
ejpam-5587	142	8	.	.	PUNCT
ejpam-5587	142	9	(	(	PUNCT
ejpam-5587	142	10	iv	iv	X
ejpam-5587	142	11	)	)	PUNCT
ejpam-5587	142	12	s0	s0	NOUN
ejpam-5587	142	13	=	=	SYM
ejpam-5587	142	14	t	t	PROPN
ejpam-5587	142	15	|d(t	|d(t	ADJ
ejpam-5587	142	16	)	)	PUNCT
ejpam-5587	142	17	∩ker(t	∩ker(t	NUM
ejpam-5587	142	18	)	)	PUNCT
ejpam-5587	143	1	⊥	⊥	NOUN
ejpam-5587	143	2	has	have	VERB
ejpam-5587	143	3	a	a	DET
ejpam-5587	143	4	bounded	bounded	ADJ
ejpam-5587	143	5	inverse	inverse	NOUN
ejpam-5587	143	6	.	.	PUNCT
ejpam-5587	144	1	if	if	SCONJ
ejpam-5587	144	2	t	t	PROPN
ejpam-5587	144	3	∈	∈	PROPN
ejpam-5587	144	4	l(h	l(h	PROPN
ejpam-5587	144	5	)	)	PUNCT
ejpam-5587	144	6	is	be	AUX
ejpam-5587	144	7	a	a	DET
ejpam-5587	144	8	densely	densely	ADV
ejpam-5587	144	9	defined	define	VERB
ejpam-5587	144	10	closed	closed	ADJ
ejpam-5587	144	11	operator	operator	NOUN
ejpam-5587	144	12	and	and	CCONJ
ejpam-5587	144	13	ker(t	ker(t	NOUN
ejpam-5587	144	14	)	)	PUNCT
ejpam-5587	145	1	=	=	PUNCT
ejpam-5587	145	2	{	{	PUNCT
ejpam-5587	145	3	0	0	NUM
ejpam-5587	145	4	}	}	PUNCT
ejpam-5587	145	5	,	,	PUNCT
ejpam-5587	145	6	then	then	ADV
ejpam-5587	145	7	the	the	DET
ejpam-5587	145	8	inverse	inverse	NOUN
ejpam-5587	145	9	operator	operator	NOUN
ejpam-5587	145	10	,	,	PUNCT
ejpam-5587	145	11	t−1	t−1	PROPN
ejpam-5587	145	12	is	be	AUX
ejpam-5587	145	13	the	the	DET
ejpam-5587	145	14	linear	linear	ADJ
ejpam-5587	145	15	operator	operator	NOUN
ejpam-5587	145	16	from	from	ADP
ejpam-5587	145	17	h	h	NOUN
ejpam-5587	145	18	to	to	ADP
ejpam-5587	145	19	h	h	NOUN
ejpam-5587	145	20	,	,	PUNCT
ejpam-5587	145	21	with	with	ADP
ejpam-5587	145	22	d(t−1	d(t−1	PROPN
ejpam-5587	145	23	)	)	PUNCT
ejpam-5587	145	24	=	=	SYM
ejpam-5587	145	25	ran(t	ran(t	PROPN
ejpam-5587	145	26	)	)	PUNCT
ejpam-5587	145	27	and	and	CCONJ
ejpam-5587	145	28	t−1tx	t−1tx	PRON
ejpam-5587	145	29	=	=	PUNCT
ejpam-5587	146	1	x	x	X
ejpam-5587	146	2	for	for	ADP
ejpam-5587	146	3	all	all	DET
ejpam-5587	146	4	x	x	SYM
ejpam-5587	146	5	∈	∈	PROPN
ejpam-5587	146	6	d(t	d(t	PROPN
ejpam-5587	146	7	)	)	PUNCT
ejpam-5587	146	8	.	.	PUNCT
ejpam-5587	147	1	in	in	ADP
ejpam-5587	147	2	particular	particular	ADJ
ejpam-5587	147	3	if	if	SCONJ
ejpam-5587	147	4	t	t	PROPN
ejpam-5587	147	5	is	be	AUX
ejpam-5587	147	6	a	a	DET
ejpam-5587	147	7	bijection	bijection	NOUN
ejpam-5587	147	8	,	,	PUNCT
ejpam-5587	147	9	then	then	ADV
ejpam-5587	147	10	by	by	ADP
ejpam-5587	147	11	the	the	DET
ejpam-5587	147	12	closed	closed	ADJ
ejpam-5587	147	13	graph	graph	NOUN
ejpam-5587	147	14	theorem	theorem	NOUN
ejpam-5587	147	15	it	it	PRON
ejpam-5587	147	16	follows	follow	VERB
ejpam-5587	147	17	that	that	SCONJ
ejpam-5587	147	18	t−1	t−1	PROPN
ejpam-5587	147	19	∈	∈	PROPN
ejpam-5587	147	20	b(h	b(h	PROPN
ejpam-5587	147	21	)	)	PUNCT
ejpam-5587	147	22	.	.	PUNCT
ejpam-5587	148	1	in	in	ADP
ejpam-5587	148	2	addition	addition	NOUN
ejpam-5587	148	3	,	,	PUNCT
ejpam-5587	148	4	if	if	SCONJ
ejpam-5587	148	5	t	t	PROPN
ejpam-5587	148	6	is	be	AUX
ejpam-5587	148	7	normal	normal	ADJ
ejpam-5587	148	8	then	then	ADV
ejpam-5587	148	9	t	t	PROPN
ejpam-5587	148	10	has	have	VERB
ejpam-5587	148	11	a	a	DET
ejpam-5587	148	12	bounded	bounded	ADJ
ejpam-5587	148	13	inverse	inverse	NOUN
ejpam-5587	148	14	if	if	SCONJ
ejpam-5587	149	1	and	and	CCONJ
ejpam-5587	149	2	only	only	ADV
ejpam-5587	149	3	if	if	SCONJ
ejpam-5587	149	4	m(t	m(t	PROPN
ejpam-5587	149	5	)	)	PUNCT
ejpam-5587	150	1	>	>	X
ejpam-5587	150	2	0	0	X
ejpam-5587	150	3	.	.	PUNCT
ejpam-5587	150	4	theorem	theorem	NOUN
ejpam-5587	150	5	3	3	X
ejpam-5587	150	6	.	.	PUNCT
ejpam-5587	151	1	if	if	SCONJ
ejpam-5587	151	2	t	t	PROPN
ejpam-5587	151	3	∈	∈	PROPN
ejpam-5587	151	4	b(h	b(h	PROPN
ejpam-5587	151	5	)	)	PUNCT
ejpam-5587	151	6	is	be	AUX
ejpam-5587	151	7	a	a	DET
ejpam-5587	151	8	totally	totally	ADV
ejpam-5587	151	9	paranormal	paranormal	ADJ
ejpam-5587	151	10	,	,	PUNCT
ejpam-5587	151	11	then	then	ADV
ejpam-5587	151	12	(	(	PUNCT
ejpam-5587	151	13	i	i	NOUN
ejpam-5587	151	14	)	)	PUNCT
ejpam-5587	151	15	t	t	PROPN
ejpam-5587	151	16	is	be	AUX
ejpam-5587	151	17	normaloid	normaloid	PROPN
ejpam-5587	151	18	.	.	PUNCT
ejpam-5587	152	1	(	(	PUNCT
ejpam-5587	152	2	ii	ii	NOUN
ejpam-5587	152	3	)	)	PUNCT
ejpam-5587	152	4	t−1	t−1	PROPN
ejpam-5587	152	5	is	be	AUX
ejpam-5587	152	6	totally	totally	ADV
ejpam-5587	152	7	paranormal	paranormal	ADJ
ejpam-5587	152	8	,	,	PUNCT
ejpam-5587	152	9	if	if	SCONJ
ejpam-5587	152	10	t	t	PROPN
ejpam-5587	152	11	is	be	AUX
ejpam-5587	152	12	invertible	invertible	ADJ
ejpam-5587	152	13	.	.	PUNCT
ejpam-5587	153	1	(	(	PUNCT
ejpam-5587	153	2	iii	iii	X
ejpam-5587	153	3	)	)	PUNCT
ejpam-5587	153	4	t	t	PROPN
ejpam-5587	153	5	is	be	AUX
ejpam-5587	153	6	unitary	unitary	ADJ
ejpam-5587	153	7	,	,	PUNCT
ejpam-5587	153	8	if	if	SCONJ
ejpam-5587	153	9	σ(t	σ(t	PROPN
ejpam-5587	153	10	)	)	PUNCT
ejpam-5587	153	11	lies	lie	VERB
ejpam-5587	153	12	on	on	ADP
ejpam-5587	153	13	the	the	DET
ejpam-5587	153	14	unit	unit	NOUN
ejpam-5587	153	15	circle	circle	NOUN
ejpam-5587	153	16	.	.	PUNCT
ejpam-5587	154	1	s.	s.	PROPN
ejpam-5587	154	2	alnabulsi	alnabulsi	PROPN
ejpam-5587	154	3	,	,	PUNCT
ejpam-5587	154	4	m.h.m	m.h.m	PROPN
ejpam-5587	154	5	.	.	PUNCT
ejpam-5587	154	6	rashid	rashid	PROPN
ejpam-5587	154	7	/	/	SYM
ejpam-5587	154	8	eur	eur	PROPN
ejpam-5587	154	9	.	.	PUNCT
ejpam-5587	155	1	j.	j.	PROPN
ejpam-5587	155	2	pure	pure	PROPN
ejpam-5587	155	3	appl	appl	PROPN
ejpam-5587	155	4	.	.	PROPN
ejpam-5587	155	5	math	math	PROPN
ejpam-5587	155	6	,	,	PUNCT
ejpam-5587	155	7	18	18	NUM
ejpam-5587	155	8	(	(	PUNCT
ejpam-5587	155	9	1	1	NUM
ejpam-5587	155	10	)	)	PUNCT
ejpam-5587	155	11	(	(	PUNCT
ejpam-5587	155	12	2025	2025	NUM
ejpam-5587	155	13	)	)	PUNCT
ejpam-5587	155	14	,	,	PUNCT
ejpam-5587	155	15	5587	5587	NUM
ejpam-5587	155	16	6	6	NUM
ejpam-5587	155	17	of	of	ADP
ejpam-5587	155	18	20	20	NUM
ejpam-5587	155	19	3	3	NUM
ejpam-5587	155	20	.	.	PUNCT
ejpam-5587	155	21	spectral	spectral	ADJ
ejpam-5587	155	22	properties	property	NOUN
ejpam-5587	155	23	in	in	ADP
ejpam-5587	155	24	this	this	DET
ejpam-5587	155	25	section	section	NOUN
ejpam-5587	155	26	,	,	PUNCT
ejpam-5587	155	27	we	we	PRON
ejpam-5587	155	28	study	study	VERB
ejpam-5587	155	29	some	some	DET
ejpam-5587	155	30	spectral	spectral	ADJ
ejpam-5587	155	31	properties	property	NOUN
ejpam-5587	155	32	of	of	ADP
ejpam-5587	155	33	densely	densely	ADV
ejpam-5587	155	34	defined	define	VERB
ejpam-5587	155	35	closed	close	VERB
ejpam-5587	155	36	totally	totally	ADV
ejpam-5587	155	37	paranormal	paranormal	ADJ
ejpam-5587	155	38	operators	operator	NOUN
ejpam-5587	155	39	.	.	PUNCT
ejpam-5587	156	1	definition	definition	NOUN
ejpam-5587	156	2	6	6	NUM
ejpam-5587	156	3	.	.	PUNCT
ejpam-5587	157	1	a	a	DET
ejpam-5587	157	2	densely	densely	ADV
ejpam-5587	157	3	defined	define	VERB
ejpam-5587	157	4	operator	operator	NOUN
ejpam-5587	157	5	t	t	PROPN
ejpam-5587	157	6	∈	∈	PROPN
ejpam-5587	157	7	l(h	l(h	PROPN
ejpam-5587	157	8	)	)	PUNCT
ejpam-5587	157	9	is	be	AUX
ejpam-5587	157	10	said	say	VERB
ejpam-5587	157	11	to	to	PART
ejpam-5587	157	12	be	be	AUX
ejpam-5587	157	13	totally	totally	ADV
ejpam-5587	157	14	paranormal	paranormal	ADJ
ejpam-5587	157	15	if	if	SCONJ
ejpam-5587	157	16	∥(t	∥(t	VERB
ejpam-5587	157	17	−	−	ADP
ejpam-5587	157	18	λi)x∥2	λi)x∥2	NOUN
ejpam-5587	157	19	≤	≤	ADJ
ejpam-5587	157	20	∥∥(t	∥∥(t	PUNCT
ejpam-5587	157	21	−	−	NOUN
ejpam-5587	157	22	λi)2x	λi)2x	VERB
ejpam-5587	157	23	∥∥	∥∥	PUNCT
ejpam-5587	157	24	∥x∥	∥x∥	NOUN
ejpam-5587	157	25	for	for	ADP
ejpam-5587	157	26	all	all	DET
ejpam-5587	157	27	λ	λ	PROPN
ejpam-5587	157	28	∈	∈	PROPN
ejpam-5587	157	29	c	c	X
ejpam-5587	157	30	and	and	CCONJ
ejpam-5587	157	31	d((t	d((t	VERB
ejpam-5587	157	32	−	−	NOUN
ejpam-5587	157	33	λ)2	λ)2	NOUN
ejpam-5587	157	34	)	)	PUNCT
ejpam-5587	157	35	⊆	⊆	PROPN
ejpam-5587	157	36	d(t	d(t	PROPN
ejpam-5587	157	37	−	−	PROPN
ejpam-5587	157	38	λ	λ	PROPN
ejpam-5587	157	39	)	)	PUNCT
ejpam-5587	157	40	.	.	PUNCT
ejpam-5587	158	1	equivalently	equivalently	ADV
ejpam-5587	158	2	,	,	PUNCT
ejpam-5587	158	3	t	t	PROPN
ejpam-5587	158	4	is	be	AUX
ejpam-5587	158	5	totally	totally	ADV
ejpam-5587	158	6	paranormal	paranormal	ADJ
ejpam-5587	158	7	if	if	SCONJ
ejpam-5587	159	1	and	and	CCONJ
ejpam-5587	159	2	only	only	ADV
ejpam-5587	160	1	if	if	SCONJ
ejpam-5587	160	2	t	t	PROPN
ejpam-5587	160	3	−	−	PROPN
ejpam-5587	160	4	λi	λi	INTJ
ejpam-5587	160	5	is	be	AUX
ejpam-5587	160	6	paranormal	paranormal	ADJ
ejpam-5587	160	7	for	for	ADP
ejpam-5587	160	8	all	all	DET
ejpam-5587	160	9	λ	λ	PROPN
ejpam-5587	160	10	∈	∈	PROPN
ejpam-5587	160	11	c.	c.	NOUN
ejpam-5587	160	12	and	and	CCONJ
ejpam-5587	160	13	t	t	PROPN
ejpam-5587	160	14	is	be	AUX
ejpam-5587	160	15	totally	totally	ADV
ejpam-5587	160	16	∗-paranormal	∗-paranormal	ADJ
ejpam-5587	160	17	if	if	SCONJ
ejpam-5587	160	18	t	t	PROPN
ejpam-5587	160	19	−	−	PROPN
ejpam-5587	160	20	λi	λi	INTJ
ejpam-5587	160	21	is	be	AUX
ejpam-5587	160	22	∗-paranormal	∗-paranormal	ADJ
ejpam-5587	160	23	for	for	ADP
ejpam-5587	160	24	all	all	DET
ejpam-5587	160	25	λ	λ	PROPN
ejpam-5587	160	26	∈	∈	PROPN
ejpam-5587	160	27	c.	c.	NOUN
ejpam-5587	160	28	a	a	DET
ejpam-5587	160	29	densely	densely	ADV
ejpam-5587	160	30	defined	define	VERB
ejpam-5587	160	31	operator	operator	NOUN
ejpam-5587	160	32	t	t	NOUN
ejpam-5587	160	33	in	in	ADP
ejpam-5587	160	34	h	h	NOUN
ejpam-5587	160	35	is	be	AUX
ejpam-5587	160	36	said	say	VERB
ejpam-5587	160	37	to	to	PART
ejpam-5587	160	38	be	be	AUX
ejpam-5587	160	39	hyponormal	hyponormal	ADJ
ejpam-5587	160	40	if	if	SCONJ
ejpam-5587	160	41	d(t	d(t	PROPN
ejpam-5587	160	42	)	)	PUNCT
ejpam-5587	161	1	⊆	⊆	PROPN
ejpam-5587	161	2	d(t	d(t	PROPN
ejpam-5587	161	3	∗	∗	NOUN
ejpam-5587	161	4	)	)	PUNCT
ejpam-5587	161	5	and	and	CCONJ
ejpam-5587	161	6	∥t	∥t	PRON
ejpam-5587	161	7	∗x∥	∗x∥	ADJ
ejpam-5587	161	8	≤	≤	NUM
ejpam-5587	161	9	∥tx∥	∥tx∥	NOUN
ejpam-5587	161	10	for	for	ADP
ejpam-5587	161	11	x	x	PROPN
ejpam-5587	161	12	∈	∈	PROPN
ejpam-5587	161	13	d(t	d(t	PROPN
ejpam-5587	161	14	)	)	PUNCT
ejpam-5587	161	15	.	.	PUNCT
ejpam-5587	162	1	and	and	CCONJ
ejpam-5587	162	2	t	t	PROPN
ejpam-5587	162	3	is	be	AUX
ejpam-5587	162	4	cohyponormal	cohyponormal	ADJ
ejpam-5587	162	5	if	if	SCONJ
ejpam-5587	162	6	t	t	PROPN
ejpam-5587	162	7	∗	∗	NOUN
ejpam-5587	162	8	is	be	AUX
ejpam-5587	162	9	hyponormal	hyponormal	ADJ
ejpam-5587	162	10	.	.	PUNCT
ejpam-5587	163	1	proposition	proposition	NOUN
ejpam-5587	163	2	1	1	NUM
ejpam-5587	163	3	.	.	PUNCT
ejpam-5587	164	1	let	let	VERB
ejpam-5587	164	2	t	t	PROPN
ejpam-5587	164	3	∈	∈	NOUN
ejpam-5587	164	4	c(h	c(h	PROPN
ejpam-5587	164	5	)	)	PUNCT
ejpam-5587	164	6	.	.	PUNCT
ejpam-5587	165	1	then	then	ADV
ejpam-5587	165	2	(	(	PUNCT
ejpam-5587	165	3	i	i	NOUN
ejpam-5587	165	4	)	)	PUNCT
ejpam-5587	165	5	if	if	SCONJ
ejpam-5587	165	6	t	t	PROPN
ejpam-5587	165	7	is	be	AUX
ejpam-5587	165	8	closed	close	VERB
ejpam-5587	165	9	cohyponormal	cohyponormal	ADJ
ejpam-5587	165	10	and	and	CCONJ
ejpam-5587	165	11	closed	close	VERB
ejpam-5587	165	12	totally	totally	ADV
ejpam-5587	165	13	∗-paranormal	∗-paranormal	ADJ
ejpam-5587	165	14	,	,	PUNCT
ejpam-5587	165	15	then	then	ADV
ejpam-5587	165	16	t	t	PROPN
ejpam-5587	165	17	is	be	AUX
ejpam-5587	165	18	closed	close	VERB
ejpam-5587	165	19	totally	totally	ADV
ejpam-5587	165	20	paranormal	paranormal	ADJ
ejpam-5587	165	21	.	.	PUNCT
ejpam-5587	166	1	(	(	PUNCT
ejpam-5587	166	2	ii	ii	NOUN
ejpam-5587	166	3	)	)	PUNCT
ejpam-5587	166	4	if	if	SCONJ
ejpam-5587	166	5	t	t	PROPN
ejpam-5587	166	6	is	be	AUX
ejpam-5587	166	7	closed	close	VERB
ejpam-5587	166	8	hyponormal	hyponormal	ADJ
ejpam-5587	166	9	and	and	CCONJ
ejpam-5587	166	10	closed	close	VERB
ejpam-5587	166	11	totally	totally	ADV
ejpam-5587	166	12	paranormal	paranormal	ADJ
ejpam-5587	166	13	,	,	PUNCT
ejpam-5587	166	14	then	then	ADV
ejpam-5587	166	15	t	t	PROPN
ejpam-5587	166	16	is	be	AUX
ejpam-5587	166	17	closed	close	VERB
ejpam-5587	166	18	totally	totally	ADV
ejpam-5587	166	19	∗-paranormal	∗-paranormal	ADJ
ejpam-5587	166	20	.	.	PUNCT
ejpam-5587	167	1	proof	proof	NOUN
ejpam-5587	167	2	.	.	PUNCT
ejpam-5587	168	1	(	(	PUNCT
ejpam-5587	168	2	i	i	NOUN
ejpam-5587	168	3	)	)	PUNCT
ejpam-5587	168	4	assume	assume	VERB
ejpam-5587	168	5	that	that	SCONJ
ejpam-5587	168	6	t	t	PROPN
ejpam-5587	168	7	is	be	AUX
ejpam-5587	168	8	closed	close	VERB
ejpam-5587	168	9	cohyponormal	cohyponormal	ADJ
ejpam-5587	168	10	and	and	CCONJ
ejpam-5587	168	11	closed	close	VERB
ejpam-5587	168	12	totally	totally	ADV
ejpam-5587	168	13	∗-paranormal	∗-paranormal	ADJ
ejpam-5587	168	14	.	.	PUNCT
ejpam-5587	169	1	if	if	SCONJ
ejpam-5587	169	2	t	t	PROPN
ejpam-5587	169	3	is	be	AUX
ejpam-5587	169	4	closed	close	VERB
ejpam-5587	169	5	cohyponormal	cohyponormal	ADJ
ejpam-5587	169	6	,	,	PUNCT
ejpam-5587	169	7	then	then	ADV
ejpam-5587	169	8	so	so	ADV
ejpam-5587	169	9	is	be	AUX
ejpam-5587	169	10	t	t	PROPN
ejpam-5587	169	11	−	−	PROPN
ejpam-5587	169	12	λi	λi	INTJ
ejpam-5587	169	13	for	for	ADP
ejpam-5587	169	14	all	all	DET
ejpam-5587	169	15	λ	λ	PROPN
ejpam-5587	169	16	∈	∈	PROPN
ejpam-5587	169	17	c.	c.	NOUN
ejpam-5587	169	18	hence	hence	ADV
ejpam-5587	169	19	,	,	PUNCT
ejpam-5587	169	20	d((t	d((t	VERB
ejpam-5587	169	21	−	−	NOUN
ejpam-5587	169	22	λi)2	λi)2	ADJ
ejpam-5587	169	23	)	)	PUNCT
ejpam-5587	169	24	⊆	⊆	NUM
ejpam-5587	169	25	d((t	d((t	NOUN
ejpam-5587	169	26	−	−	NOUN
ejpam-5587	169	27	λi)∗	λi)∗	NUM
ejpam-5587	169	28	)	)	PUNCT
ejpam-5587	169	29	⊆	⊆	PROPN
ejpam-5587	169	30	d(t	d(t	PROPN
ejpam-5587	169	31	−	−	PROPN
ejpam-5587	169	32	λi	λi	NOUN
ejpam-5587	169	33	)	)	PUNCT
ejpam-5587	169	34	.	.	PUNCT
ejpam-5587	170	1	let	let	VERB
ejpam-5587	170	2	x	x	PUNCT
ejpam-5587	170	3	∈	∈	VERB
ejpam-5587	170	4	d((t	d((t	NOUN
ejpam-5587	170	5	−	−	NOUN
ejpam-5587	170	6	λi)2	λi)2	PROPN
ejpam-5587	170	7	)	)	PUNCT
ejpam-5587	170	8	.	.	PUNCT
ejpam-5587	171	1	then	then	ADV
ejpam-5587	171	2	we	we	PRON
ejpam-5587	171	3	have	have	AUX
ejpam-5587	171	4	∥(t	∥(t	VERB
ejpam-5587	171	5	−	−	ADP
ejpam-5587	171	6	λi)x∥2	λi)x∥2	NOUN
ejpam-5587	171	7	≤	≤	NUM
ejpam-5587	171	8	∥(t	∥(t	NOUN
ejpam-5587	171	9	−	−	X
ejpam-5587	171	10	λi)∗x∥2	λi)∗x∥2	NOUN
ejpam-5587	171	11	≤	≤	NOUN
ejpam-5587	171	12	∥∥(t	∥∥(t	PUNCT
ejpam-5587	171	13	−	−	NOUN
ejpam-5587	171	14	λi)2x	λi)2x	VERB
ejpam-5587	171	15	∥∥	∥∥	X
ejpam-5587	171	16	∥x∥	∥x∥	NOUN
ejpam-5587	171	17	.	.	PUNCT
ejpam-5587	172	1	(	(	PUNCT
ejpam-5587	172	2	ii	ii	NOUN
ejpam-5587	172	3	)	)	PUNCT
ejpam-5587	172	4	by	by	ADP
ejpam-5587	172	5	the	the	DET
ejpam-5587	172	6	hypotheses	hypothesis	NOUN
ejpam-5587	172	7	on	on	ADP
ejpam-5587	172	8	t	t	NOUN
ejpam-5587	172	9	,	,	PUNCT
ejpam-5587	172	10	d((t	d((t	VERB
ejpam-5587	172	11	−	−	NOUN
ejpam-5587	172	12	λi)2	λi)2	ADJ
ejpam-5587	172	13	)	)	PUNCT
ejpam-5587	172	14	⊆	⊆	PROPN
ejpam-5587	172	15	d(t	d(t	PROPN
ejpam-5587	172	16	−	−	PROPN
ejpam-5587	172	17	λi	λi	NOUN
ejpam-5587	172	18	)	)	PUNCT
ejpam-5587	172	19	⊆	⊆	NUM
ejpam-5587	172	20	d((t	d((t	NOUN
ejpam-5587	172	21	−	−	NOUN
ejpam-5587	172	22	λi)∗	λi)∗	NUM
ejpam-5587	172	23	)	)	PUNCT
ejpam-5587	172	24	,	,	PUNCT
ejpam-5587	172	25	and	and	CCONJ
ejpam-5587	172	26	for	for	ADP
ejpam-5587	172	27	each	each	DET
ejpam-5587	172	28	x	x	SYM
ejpam-5587	172	29	∈	∈	PROPN
ejpam-5587	172	30	d((t	d((t	NOUN
ejpam-5587	172	31	−	−	NOUN
ejpam-5587	172	32	λi)2	λi)2	PROPN
ejpam-5587	172	33	)	)	PUNCT
ejpam-5587	172	34	,	,	PUNCT
ejpam-5587	172	35	∥(t	∥(t	VERB
ejpam-5587	172	36	−	−	X
ejpam-5587	172	37	λi)∗x∥2	λi)∗x∥2	NOUN
ejpam-5587	172	38	≤	≤	NUM
ejpam-5587	172	39	∥(t	∥(t	VERB
ejpam-5587	173	1	−	−	PROPN
ejpam-5587	173	2	λi)x∥2	λi)x∥2	NOUN
ejpam-5587	173	3	≤	≤	ADJ
ejpam-5587	173	4	∥∥(t	∥∥(t	PUNCT
ejpam-5587	173	5	−	−	NOUN
ejpam-5587	173	6	λi)2x	λi)2x	VERB
ejpam-5587	173	7	∥∥	∥∥	X
ejpam-5587	173	8	∥x∥	∥x∥	NOUN
ejpam-5587	173	9	.	.	PUNCT
ejpam-5587	174	1	so	so	ADV
ejpam-5587	174	2	,	,	PUNCT
ejpam-5587	174	3	the	the	DET
ejpam-5587	174	4	proof	proof	NOUN
ejpam-5587	174	5	is	be	AUX
ejpam-5587	174	6	complete	complete	ADJ
ejpam-5587	174	7	.	.	PUNCT
ejpam-5587	175	1	proposition	proposition	NOUN
ejpam-5587	175	2	2	2	NUM
ejpam-5587	175	3	.	.	PUNCT
ejpam-5587	176	1	let	let	VERB
ejpam-5587	176	2	t	t	PROPN
ejpam-5587	176	3	∈	∈	NOUN
ejpam-5587	176	4	c(h	c(h	PROPN
ejpam-5587	176	5	)	)	PUNCT
ejpam-5587	176	6	be	be	VERB
ejpam-5587	176	7	totally	totally	ADV
ejpam-5587	176	8	∗-paranormal	∗-paranormal	ADJ
ejpam-5587	176	9	,	,	PUNCT
ejpam-5587	176	10	and	and	CCONJ
ejpam-5587	176	11	let	let	VERB
ejpam-5587	176	12	λ	λ	NOUN
ejpam-5587	176	13	be	be	AUX
ejpam-5587	176	14	any	any	DET
ejpam-5587	176	15	complex	complex	ADJ
ejpam-5587	176	16	scalar	scalar	NOUN
ejpam-5587	176	17	.	.	PUNCT
ejpam-5587	177	1	then	then	ADV
ejpam-5587	177	2	ker(t	ker(t	NOUN
ejpam-5587	177	3	−	−	PROPN
ejpam-5587	177	4	λi	λi	NOUN
ejpam-5587	177	5	)	)	PUNCT
ejpam-5587	177	6	⊂	⊂	PROPN
ejpam-5587	178	1	ker(t	ker(t	PROPN
ejpam-5587	178	2	−	−	NUM
ejpam-5587	179	1	λi)∗.	λi)∗.	NOUN
ejpam-5587	179	2	proof	proof	NOUN
ejpam-5587	179	3	.	.	PUNCT
ejpam-5587	180	1	let	let	VERB
ejpam-5587	180	2	x	x	SYM
ejpam-5587	180	3	∈	∈	PROPN
ejpam-5587	180	4	d(t	d(t	PROPN
ejpam-5587	180	5	)	)	PUNCT
ejpam-5587	180	6	be	be	AUX
ejpam-5587	180	7	a	a	DET
ejpam-5587	180	8	unit	unit	NOUN
ejpam-5587	180	9	eigenvector	eigenvector	NOUN
ejpam-5587	180	10	of	of	ADP
ejpam-5587	180	11	t	t	PROPN
ejpam-5587	180	12	associated	associate	VERB
ejpam-5587	180	13	to	to	ADP
ejpam-5587	180	14	λ	λ	PROPN
ejpam-5587	180	15	.	.	PUNCT
ejpam-5587	181	1	then	then	ADV
ejpam-5587	181	2	tx	tx	PROPN
ejpam-5587	181	3	=	=	SYM
ejpam-5587	181	4	λx	λx	PROPN
ejpam-5587	181	5	.	.	PUNCT
ejpam-5587	182	1	by	by	ADP
ejpam-5587	182	2	the	the	DET
ejpam-5587	182	3	hypotheses	hypothesis	NOUN
ejpam-5587	182	4	on	on	ADP
ejpam-5587	182	5	t	t	PROPN
ejpam-5587	182	6	,	,	PUNCT
ejpam-5587	182	7	∥t	∥t	PROPN
ejpam-5587	182	8	∗x∥	∗x∥	ADJ
ejpam-5587	182	9	≤	≤	NUM
ejpam-5587	182	10	|λ|	|λ|	NOUN
ejpam-5587	182	11	.	.	PUNCT
ejpam-5587	183	1	consequently	consequently	ADV
ejpam-5587	183	2	,	,	PUNCT
ejpam-5587	183	3	0	0	NUM
ejpam-5587	183	4	≤	≤	NUM
ejpam-5587	183	5	∥∥t	∥∥t	VERB
ejpam-5587	183	6	∗x−	∗x−	NOUN
ejpam-5587	183	7	λ̄x	λ̄x	NUM
ejpam-5587	183	8	∥∥2	∥∥2	PUNCT
ejpam-5587	184	1	=	=	SYM
ejpam-5587	184	2	∥t	∥t	ADJ
ejpam-5587	184	3	∗x∥2	∗x∥2	NOUN
ejpam-5587	184	4	−	−	NOUN
ejpam-5587	184	5	λ	λ	NOUN
ejpam-5587	184	6	⟨x	⟨x	VERB
ejpam-5587	184	7	,	,	PUNCT
ejpam-5587	184	8	tx⟩	tx⟩	PROPN
ejpam-5587	185	1	−	−	PROPN
ejpam-5587	185	2	λ̄	λ̄	PROPN
ejpam-5587	185	3	⟨tx	⟨tx	PROPN
ejpam-5587	185	4	,	,	PUNCT
ejpam-5587	185	5	x⟩+	x⟩+	NOUN
ejpam-5587	185	6	|λ|2	|λ|2	PROPN
ejpam-5587	185	7	=	=	SYM
ejpam-5587	185	8	∥t	∥t	ADJ
ejpam-5587	185	9	∗x∥2	∗x∥2	NOUN
ejpam-5587	185	10	−	−	ADP
ejpam-5587	185	11	|λ|2	|λ|2	PROPN
ejpam-5587	185	12	≤	≤	NOUN
ejpam-5587	185	13	0	0	NUM
ejpam-5587	185	14	.	.	PUNCT
ejpam-5587	186	1	thus	thus	ADV
ejpam-5587	186	2	,	,	PUNCT
ejpam-5587	186	3	t	t	PROPN
ejpam-5587	186	4	∗x	∗x	PROPN
ejpam-5587	186	5	=	=	PUNCT
ejpam-5587	187	1	λ̄x	λ̄x	VERB
ejpam-5587	187	2	and	and	CCONJ
ejpam-5587	187	3	so	so	ADV
ejpam-5587	187	4	x	x	SYM
ejpam-5587	187	5	∈	∈	NOUN
ejpam-5587	187	6	ker(t	ker(t	NOUN
ejpam-5587	187	7	−	−	PROPN
ejpam-5587	187	8	λ)∗.	λ)∗.	PROPN
ejpam-5587	187	9	s.	s.	PROPN
ejpam-5587	187	10	alnabulsi	alnabulsi	PROPN
ejpam-5587	187	11	,	,	PUNCT
ejpam-5587	187	12	m.h.m	m.h.m	PROPN
ejpam-5587	187	13	.	.	PUNCT
ejpam-5587	187	14	rashid	rashid	PROPN
ejpam-5587	187	15	/	/	SYM
ejpam-5587	187	16	eur	eur	PROPN
ejpam-5587	187	17	.	.	PUNCT
ejpam-5587	188	1	j.	j.	PROPN
ejpam-5587	188	2	pure	pure	PROPN
ejpam-5587	188	3	appl	appl	PROPN
ejpam-5587	188	4	.	.	PROPN
ejpam-5587	188	5	math	math	PROPN
ejpam-5587	188	6	,	,	PUNCT
ejpam-5587	188	7	18	18	NUM
ejpam-5587	188	8	(	(	PUNCT
ejpam-5587	188	9	1	1	NUM
ejpam-5587	188	10	)	)	PUNCT
ejpam-5587	188	11	(	(	PUNCT
ejpam-5587	188	12	2025	2025	NUM
ejpam-5587	188	13	)	)	PUNCT
ejpam-5587	188	14	,	,	PUNCT
ejpam-5587	188	15	5587	5587	NUM
ejpam-5587	188	16	7	7	NUM
ejpam-5587	188	17	of	of	ADP
ejpam-5587	188	18	20	20	NUM
ejpam-5587	188	19	lemma	lemma	PROPN
ejpam-5587	188	20	2	2	NUM
ejpam-5587	188	21	.	.	PUNCT
ejpam-5587	189	1	let	let	AUX
ejpam-5587	189	2	t	t	PROPN
ejpam-5587	189	3	∈	∈	NOUN
ejpam-5587	189	4	c(h	c(h	PROPN
ejpam-5587	189	5	)	)	PUNCT
ejpam-5587	189	6	be	be	AUX
ejpam-5587	189	7	a	a	DET
ejpam-5587	189	8	totally	totally	ADV
ejpam-5587	189	9	∗-paranormal	∗-paranormal	ADJ
ejpam-5587	189	10	,	,	PUNCT
ejpam-5587	189	11	then	then	ADV
ejpam-5587	189	12	there	there	PRON
ejpam-5587	189	13	exists	exist	VERB
ejpam-5587	189	14	a	a	DET
ejpam-5587	189	15	contraction	contraction	NOUN
ejpam-5587	189	16	qλ	qλ	PROPN
ejpam-5587	189	17	∈	∈	PROPN
ejpam-5587	189	18	b(h	b(h	PROPN
ejpam-5587	189	19	)	)	PUNCT
ejpam-5587	189	20	such	such	ADJ
ejpam-5587	189	21	that	that	SCONJ
ejpam-5587	189	22	(	(	PUNCT
ejpam-5587	189	23	t	t	PROPN
ejpam-5587	189	24	−	−	PROPN
ejpam-5587	189	25	λi)2	λi)2	PROPN
ejpam-5587	189	26	⊂	⊂	PROPN
ejpam-5587	189	27	(	(	PUNCT
ejpam-5587	189	28	t	t	NOUN
ejpam-5587	189	29	−	−	NOUN
ejpam-5587	189	30	λi)∗qλ	λi)∗qλ	NOUN
ejpam-5587	189	31	.	.	PUNCT
ejpam-5587	190	1	proof	proof	NOUN
ejpam-5587	190	2	.	.	PUNCT
ejpam-5587	191	1	since	since	SCONJ
ejpam-5587	191	2	∥(t	∥(t	NOUN
ejpam-5587	191	3	−	−	PROPN
ejpam-5587	191	4	λi)∗x∥2	λi)∗x∥2	NOUN
ejpam-5587	191	5	≤	≤	NOUN
ejpam-5587	191	6	∥∥(t	∥∥(t	PUNCT
ejpam-5587	191	7	−	−	NOUN
ejpam-5587	191	8	λi)2x	λi)2x	VERB
ejpam-5587	191	9	∥∥	∥∥	PUNCT
ejpam-5587	191	10	∥x∥	∥x∥	NOUN
ejpam-5587	191	11	for	for	ADP
ejpam-5587	191	12	all	all	DET
ejpam-5587	191	13	x	x	SYM
ejpam-5587	191	14	∈	∈	NOUN
ejpam-5587	191	15	d((t	d((t	NOUN
ejpam-5587	192	1	−	−	NOUN
ejpam-5587	192	2	λi)2	λi)2	ADJ
ejpam-5587	192	3	)	)	PUNCT
ejpam-5587	192	4	⊆	⊆	NUM
ejpam-5587	192	5	d((t	d((t	NOUN
ejpam-5587	192	6	−	−	NOUN
ejpam-5587	192	7	λi)∗	λi)∗	NUM
ejpam-5587	192	8	)	)	PUNCT
ejpam-5587	193	1	,	,	PUNCT
ejpam-5587	193	2	there	there	PRON
ejpam-5587	193	3	exists	exist	VERB
ejpam-5587	193	4	a	a	DET
ejpam-5587	193	5	contraction	contraction	NOUN
ejpam-5587	193	6	k	k	NOUN
ejpam-5587	193	7	′	′	NUM
ejpam-5587	193	8	∈	∈	PROPN
ejpam-5587	193	9	b	b	PROPN
ejpam-5587	193	10	(	(	PUNCT
ejpam-5587	193	11	ran	run	VERB
ejpam-5587	193	12	(	(	PUNCT
ejpam-5587	193	13	(	(	PUNCT
ejpam-5587	193	14	t	t	PROPN
ejpam-5587	193	15	−	−	PROPN
ejpam-5587	193	16	λi)2	λi)2	PROPN
ejpam-5587	193	17	)	)	PUNCT
ejpam-5587	193	18	,	,	PUNCT
ejpam-5587	193	19	ran(t	ran(t	X
ejpam-5587	193	20	∗	∗	NOUN
ejpam-5587	193	21	−	−	PROPN
ejpam-5587	193	22	λ̄i	λ̄i	NOUN
ejpam-5587	193	23	)	)	PUNCT
ejpam-5587	193	24	)	)	PUNCT
ejpam-5587	193	25	such	such	ADJ
ejpam-5587	193	26	that	that	SCONJ
ejpam-5587	193	27	k	k	PROPN
ejpam-5587	194	1	′(t−	′(t−	PROPN
ejpam-5587	195	1	λi)2	λi)2	PROPN
ejpam-5587	195	2	⊂	⊂	PROPN
ejpam-5587	195	3	t	t	PROPN
ejpam-5587	195	4	∗	∗	NOUN
ejpam-5587	195	5	−	−	PROPN
ejpam-5587	196	1	λ̄i	λ̄i	PROPN
ejpam-5587	196	2	.	.	PUNCT
ejpam-5587	197	1	let	let	VERB
ejpam-5587	197	2	k	k	PROPN
ejpam-5587	197	3	∈	∈	PROPN
ejpam-5587	197	4	b(h	b(h	PROPN
ejpam-5587	197	5	)	)	PUNCT
ejpam-5587	197	6	be	be	VERB
ejpam-5587	197	7	any	any	DET
ejpam-5587	197	8	contraction	contraction	NOUN
ejpam-5587	197	9	which	which	PRON
ejpam-5587	197	10	extends	extend	VERB
ejpam-5587	197	11	k	k	PROPN
ejpam-5587	197	12	′	′	NUM
ejpam-5587	197	13	(	(	PUNCT
ejpam-5587	197	14	e.g.	e.g.	ADV
ejpam-5587	197	15	set	set	VERB
ejpam-5587	197	16	kx	kx	PROPN
ejpam-5587	197	17	=	=	NOUN
ejpam-5587	197	18	0	0	PROPN
ejpam-5587	197	19	for	for	ADP
ejpam-5587	197	20	x	x	PROPN
ejpam-5587	197	21	∈	∈	PROPN
ejpam-5587	197	22	h⊖ran	h⊖ran	NOUN
ejpam-5587	197	23	(	(	PUNCT
ejpam-5587	197	24	(	(	PUNCT
ejpam-5587	197	25	t	t	PROPN
ejpam-5587	197	26	−	−	PROPN
ejpam-5587	197	27	λi)2	λi)2	PROPN
ejpam-5587	197	28	)	)	PUNCT
ejpam-5587	197	29	)	)	PUNCT
ejpam-5587	197	30	.	.	PUNCT
ejpam-5587	198	1	thenk(t−λi)2	thenk(t−λi)2	PROPN
ejpam-5587	198	2	⊂	⊂	PROPN
ejpam-5587	198	3	t	t	PROPN
ejpam-5587	198	4	∗−λ̄i	∗−λ̄i	PROPN
ejpam-5587	198	5	.	.	PUNCT
ejpam-5587	199	1	taking	take	VERB
ejpam-5587	199	2	adjoints	adjoint	NOUN
ejpam-5587	199	3	in	in	ADP
ejpam-5587	199	4	the	the	DET
ejpam-5587	199	5	last	last	ADJ
ejpam-5587	199	6	inclusion	inclusion	NOUN
ejpam-5587	199	7	and	and	CCONJ
ejpam-5587	199	8	exploiting	exploit	VERB
ejpam-5587	199	9	the	the	DET
ejpam-5587	199	10	closability	closability	NOUN
ejpam-5587	199	11	of	of	ADP
ejpam-5587	199	12	t	t	PROPN
ejpam-5587	199	13	,	,	PUNCT
ejpam-5587	199	14	we	we	PRON
ejpam-5587	199	15	get	get	VERB
ejpam-5587	199	16	(	(	PUNCT
ejpam-5587	199	17	t	t	NOUN
ejpam-5587	199	18	−λi)2	−λi)2	PROPN
ejpam-5587	199	19	⊆	⊆	NUM
ejpam-5587	199	20	(	(	PUNCT
ejpam-5587	199	21	(	(	PUNCT
ejpam-5587	199	22	t	t	NOUN
ejpam-5587	199	23	−	−	PROPN
ejpam-5587	199	24	λi)2	λi)2	PROPN
ejpam-5587	199	25	)	)	PUNCT
ejpam-5587	200	1	∗∗	∗∗	PROPN
ejpam-5587	200	2	⊆	⊆	NUM
ejpam-5587	200	3	(	(	PUNCT
ejpam-5587	200	4	k(t	k(t	PROPN
ejpam-5587	200	5	−	−	PROPN
ejpam-5587	200	6	λi)2	λi)2	ADJ
ejpam-5587	200	7	)	)	PUNCT
ejpam-5587	200	8	∗	∗	NOUN
ejpam-5587	200	9	=	=	SYM
ejpam-5587	200	10	(	(	PUNCT
ejpam-5587	200	11	t	t	PROPN
ejpam-5587	200	12	−	−	PROPN
ejpam-5587	200	13	λi)∗2k∗	λi)∗2k∗	PROPN
ejpam-5587	200	14	⊆	⊆	NUM
ejpam-5587	200	15	(	(	PUNCT
ejpam-5587	200	16	t	t	NOUN
ejpam-5587	200	17	−	−	PROPN
ejpam-5587	201	1	λi)∗k∗.	λi)∗k∗.	ADV
ejpam-5587	201	2	this	this	PRON
ejpam-5587	201	3	gives	give	VERB
ejpam-5587	201	4	us	we	PRON
ejpam-5587	201	5	the	the	DET
ejpam-5587	201	6	conclusion	conclusion	NOUN
ejpam-5587	201	7	with	with	ADP
ejpam-5587	201	8	q	q	PROPN
ejpam-5587	201	9	=	=	PUNCT
ejpam-5587	201	10	k∗.	k∗.	X
ejpam-5587	201	11	a	a	DET
ejpam-5587	201	12	closed	closed	ADJ
ejpam-5587	201	13	subspace	subspace	NOUN
ejpam-5587	201	14	m	m	NOUN
ejpam-5587	201	15	of	of	ADP
ejpam-5587	201	16	h	h	NOUN
ejpam-5587	201	17	reduces	reduce	VERB
ejpam-5587	201	18	t	t	PROPN
ejpam-5587	201	19	∈	∈	NOUN
ejpam-5587	201	20	c(h	c(h	PROPN
ejpam-5587	201	21	)	)	PUNCT
ejpam-5587	201	22	if	if	SCONJ
ejpam-5587	201	23	m	m	VERB
ejpam-5587	201	24	and	and	CCONJ
ejpam-5587	201	25	m⊥	m⊥	NOUN
ejpam-5587	201	26	are	be	AUX
ejpam-5587	201	27	invariant	invariant	ADJ
ejpam-5587	201	28	under	under	ADP
ejpam-5587	201	29	t	t	PROPN
ejpam-5587	201	30	.	.	PUNCT
ejpam-5587	202	1	stochel	stochel	PROPN
ejpam-5587	203	1	[	[	X
ejpam-5587	203	2	23	23	NUM
ejpam-5587	203	3	]	]	PUNCT
ejpam-5587	203	4	proved	prove	VERB
ejpam-5587	203	5	if	if	SCONJ
ejpam-5587	203	6	t	t	PROPN
ejpam-5587	203	7	∈	∈	NOUN
ejpam-5587	203	8	c(h	c(h	PROPN
ejpam-5587	203	9	)	)	PUNCT
ejpam-5587	203	10	is	be	AUX
ejpam-5587	203	11	hyponormal	hyponormal	ADJ
ejpam-5587	203	12	and	and	CCONJ
ejpam-5587	203	13	m	m	PROPN
ejpam-5587	203	14	is	be	AUX
ejpam-5587	203	15	a	a	DET
ejpam-5587	203	16	closed	closed	ADJ
ejpam-5587	203	17	subspace	subspace	NOUN
ejpam-5587	203	18	of	of	ADP
ejpam-5587	203	19	h	h	PRON
ejpam-5587	203	20	which	which	PRON
ejpam-5587	203	21	is	be	AUX
ejpam-5587	203	22	invariant	invariant	ADJ
ejpam-5587	203	23	under	under	ADP
ejpam-5587	203	24	t	t	PROPN
ejpam-5587	203	25	with	with	ADP
ejpam-5587	203	26	t	t	PROPN
ejpam-5587	203	27	|m	|m	NOUN
ejpam-5587	203	28	is	be	AUX
ejpam-5587	203	29	normal	normal	ADJ
ejpam-5587	203	30	,	,	PUNCT
ejpam-5587	203	31	then	then	ADV
ejpam-5587	203	32	m	m	VERB
ejpam-5587	203	33	reduces	reduce	VERB
ejpam-5587	203	34	t	t	PROPN
ejpam-5587	203	35	.	.	PUNCT
ejpam-5587	204	1	theorem	theorem	ADJ
ejpam-5587	204	2	4	4	NUM
ejpam-5587	204	3	.	.	PUNCT
ejpam-5587	205	1	suppose	suppose	VERB
ejpam-5587	205	2	t	t	PROPN
ejpam-5587	205	3	∈	∈	PROPN
ejpam-5587	205	4	c(h	c(h	PROPN
ejpam-5587	205	5	)	)	PUNCT
ejpam-5587	205	6	is	be	AUX
ejpam-5587	205	7	a	a	DET
ejpam-5587	205	8	densely	densely	ADV
ejpam-5587	205	9	defined	define	VERB
ejpam-5587	205	10	totally	totally	ADV
ejpam-5587	205	11	∗-paranormal	∗-paranormal	ADJ
ejpam-5587	205	12	.	.	PUNCT
ejpam-5587	206	1	if	if	SCONJ
ejpam-5587	206	2	m	m	NOUN
ejpam-5587	206	3	is	be	AUX
ejpam-5587	206	4	a	a	DET
ejpam-5587	206	5	closed	closed	ADJ
ejpam-5587	206	6	subspace	subspace	NOUN
ejpam-5587	206	7	of	of	ADP
ejpam-5587	206	8	h	h	PRON
ejpam-5587	206	9	which	which	PRON
ejpam-5587	206	10	is	be	AUX
ejpam-5587	206	11	invariant	invariant	ADJ
ejpam-5587	206	12	under	under	ADP
ejpam-5587	206	13	t	t	PROPN
ejpam-5587	206	14	with	with	ADP
ejpam-5587	206	15	t	t	PROPN
ejpam-5587	206	16	|m	|m	NOUN
ejpam-5587	206	17	is	be	AUX
ejpam-5587	206	18	normal	normal	ADJ
ejpam-5587	206	19	,	,	PUNCT
ejpam-5587	206	20	then	then	ADV
ejpam-5587	206	21	m	m	VERB
ejpam-5587	206	22	reduces	reduce	VERB
ejpam-5587	206	23	t	t	NOUN
ejpam-5587	206	24	.	.	PUNCT
ejpam-5587	207	1	proof	proof	NOUN
ejpam-5587	207	2	.	.	PUNCT
ejpam-5587	208	1	let	let	VERB
ejpam-5587	208	2	h	h	NOUN
ejpam-5587	208	3	=	=	SYM
ejpam-5587	208	4	h1⊕h2	h1⊕h2	PROPN
ejpam-5587	208	5	,	,	PUNCT
ejpam-5587	208	6	where	where	SCONJ
ejpam-5587	208	7	h1	h1	VERB
ejpam-5587	208	8	=	=	SYM
ejpam-5587	208	9	m	m	PROPN
ejpam-5587	208	10	and	and	CCONJ
ejpam-5587	208	11	h2	h2	NOUN
ejpam-5587	208	12	=	=	PRON
ejpam-5587	208	13	m⊥.	m⊥.	ADV
ejpam-5587	208	14	then	then	ADV
ejpam-5587	208	15	t	t	PROPN
ejpam-5587	208	16	has	have	VERB
ejpam-5587	208	17	the	the	DET
ejpam-5587	208	18	block	block	NOUN
ejpam-5587	208	19	matrix	matrix	NOUN
ejpam-5587	208	20	representation	representation	NOUN
ejpam-5587	208	21	[	[	PUNCT
ejpam-5587	208	22	t11	t11	PROPN
ejpam-5587	208	23	t12	t12	PROPN
ejpam-5587	208	24	t21	t21	PROPN
ejpam-5587	208	25	t22	t22	PROPN
ejpam-5587	208	26	]	]	PUNCT
ejpam-5587	208	27	,	,	PUNCT
ejpam-5587	208	28	where	where	SCONJ
ejpam-5587	208	29	tij	tij	NOUN
ejpam-5587	208	30	:	:	PUNCT
ejpam-5587	208	31	d(t	d(t	PROPN
ejpam-5587	208	32	)	)	PUNCT
ejpam-5587	208	33	∩	∩	PROPN
ejpam-5587	208	34	hj	hj	PROPN
ejpam-5587	208	35	→	→	AUX
ejpam-5587	208	36	hi	hi	PROPN
ejpam-5587	208	37	is	be	AUX
ejpam-5587	208	38	defined	define	VERB
ejpam-5587	208	39	by	by	ADP
ejpam-5587	208	40	tij	tij	PROPN
ejpam-5587	208	41	=	=	SYM
ejpam-5587	208	42	p	p	PROPN
ejpam-5587	208	43	|hitp	|hitp	PROPN
ejpam-5587	208	44	|hj	|hj	X
ejpam-5587	208	45	|d(t	|d(t	ADJ
ejpam-5587	208	46	)	)	PUNCT
ejpam-5587	208	47	∩hk	∩hk	NOUN
ejpam-5587	208	48	for	for	ADP
ejpam-5587	208	49	k	k	PROPN
ejpam-5587	208	50	=	=	SYM
ejpam-5587	208	51	1	1	NUM
ejpam-5587	208	52	,	,	PUNCT
ejpam-5587	208	53	2	2	NUM
ejpam-5587	208	54	.	.	PUNCT
ejpam-5587	209	1	here	here	ADV
ejpam-5587	209	2	,	,	PUNCT
ejpam-5587	209	3	p	p	PROPN
ejpam-5587	209	4	|hi	|hi	PRON
ejpam-5587	209	5	denotes	denote	VERB
ejpam-5587	209	6	the	the	DET
ejpam-5587	209	7	orthogonal	orthogonal	ADJ
ejpam-5587	209	8	projection	projection	NOUN
ejpam-5587	209	9	onto	onto	ADP
ejpam-5587	209	10	hi	hi	INTJ
ejpam-5587	209	11	.	.	PUNCT
ejpam-5587	210	1	since	since	SCONJ
ejpam-5587	210	2	m	m	PROPN
ejpam-5587	210	3	is	be	AUX
ejpam-5587	210	4	invariant	invariant	ADJ
ejpam-5587	210	5	under	under	ADP
ejpam-5587	210	6	t	t	PROPN
ejpam-5587	210	7	,	,	PUNCT
ejpam-5587	210	8	we	we	PRON
ejpam-5587	210	9	have	have	VERB
ejpam-5587	210	10	[	[	PUNCT
ejpam-5587	210	11	t11	t11	PROPN
ejpam-5587	210	12	t12	t12	PROPN
ejpam-5587	210	13	0	0	NUM
ejpam-5587	210	14	t22	t22	PROPN
ejpam-5587	210	15	]	]	PUNCT
ejpam-5587	210	16	.	.	PUNCT
ejpam-5587	211	1	let	let	VERB
ejpam-5587	211	2	y	y	PROPN
ejpam-5587	211	3	∈	∈	PROPN
ejpam-5587	211	4	d(t	d(t	PROPN
ejpam-5587	211	5	)	)	PUNCT
ejpam-5587	212	1	∩m⊥.	∩m⊥.	PROPN
ejpam-5587	212	2	by	by	ADP
ejpam-5587	212	3	lemma	lemma	PROPN
ejpam-5587	212	4	2	2	NUM
ejpam-5587	212	5	,	,	PUNCT
ejpam-5587	212	6	we	we	PRON
ejpam-5587	212	7	have	have	VERB
ejpam-5587	212	8	(	(	PUNCT
ejpam-5587	212	9	t	t	NOUN
ejpam-5587	212	10	−	−	PROPN
ejpam-5587	212	11	λi)2	λi)2	PROPN
ejpam-5587	212	12	⊂	⊂	PROPN
ejpam-5587	212	13	(	(	PUNCT
ejpam-5587	212	14	t	t	PROPN
ejpam-5587	212	15	−	−	NOUN
ejpam-5587	212	16	λi)∗qλ	λi)∗qλ	NOUN
ejpam-5587	212	17	for	for	ADP
ejpam-5587	212	18	every	every	DET
ejpam-5587	212	19	λ	λ	PROPN
ejpam-5587	212	20	∈	∈	PROPN
ejpam-5587	212	21	c.	c.	NOUN
ejpam-5587	212	22	thus	thus	ADV
ejpam-5587	212	23	,	,	PUNCT
ejpam-5587	212	24	ran	run	VERB
ejpam-5587	212	25	(	(	PUNCT
ejpam-5587	212	26	(	(	PUNCT
ejpam-5587	212	27	t	t	NOUN
ejpam-5587	212	28	−	−	PROPN
ejpam-5587	212	29	λi)2	λi)2	PROPN
ejpam-5587	212	30	)	)	PUNCT
ejpam-5587	213	1	⊆	⊆	NUM
ejpam-5587	213	2	ran	run	VERB
ejpam-5587	213	3	(	(	PUNCT
ejpam-5587	213	4	(	(	PUNCT
ejpam-5587	213	5	t	t	PROPN
ejpam-5587	213	6	−	−	PROPN
ejpam-5587	213	7	λi)∗)for	λi)∗)for	ADP
ejpam-5587	213	8	every	every	DET
ejpam-5587	213	9	λ	λ	PROPN
ejpam-5587	213	10	∈	∈	PROPN
ejpam-5587	213	11	c.	c.	NOUN
ejpam-5587	213	12	then	then	ADV
ejpam-5587	213	13	there	there	PRON
ejpam-5587	213	14	exist	exist	VERB
ejpam-5587	213	15	a	a	DET
ejpam-5587	213	16	densely	densely	ADV
ejpam-5587	213	17	defined	define	VERB
ejpam-5587	213	18	operator	operator	NOUN
ejpam-5587	213	19	b	b	NOUN
ejpam-5587	213	20	such	such	ADJ
ejpam-5587	213	21	that	that	PRON
ejpam-5587	213	22	(	(	PUNCT
ejpam-5587	213	23	t	t	NOUN
ejpam-5587	213	24	−	−	NOUN
ejpam-5587	213	25	λi)2	λi)2	PROPN
ejpam-5587	213	26	=	=	SYM
ejpam-5587	213	27	(	(	PUNCT
ejpam-5587	213	28	t	t	PROPN
ejpam-5587	213	29	−	−	PROPN
ejpam-5587	213	30	λi)∗b	λi)∗b	X
ejpam-5587	213	31	(	(	PUNCT
ejpam-5587	213	32	see	see	VERB
ejpam-5587	213	33	[	[	X
ejpam-5587	213	34	7	7	NUM
ejpam-5587	213	35	]	]	NUM
ejpam-5587	213	36	)	)	PUNCT
ejpam-5587	213	37	.	.	PUNCT
ejpam-5587	214	1	hence	hence	ADV
ejpam-5587	214	2	,	,	PUNCT
ejpam-5587	214	3	t12(y	t12(y	PRON
ejpam-5587	214	4	)	)	PUNCT
ejpam-5587	214	5	=	=	PUNCT
ejpam-5587	215	1	(	(	PUNCT
ejpam-5587	215	2	t11	t11	NOUN
ejpam-5587	215	3	−	−	NOUN
ejpam-5587	215	4	λi)∗u	λi)∗u	ADP
ejpam-5587	215	5	for	for	ADP
ejpam-5587	215	6	some	some	DET
ejpam-5587	215	7	u	u	PROPN
ejpam-5587	215	8	∈	∈	PROPN
ejpam-5587	215	9	m.	m.	NOUN
ejpam-5587	215	10	we	we	PRON
ejpam-5587	215	11	can	can	AUX
ejpam-5587	215	12	choose	choose	VERB
ejpam-5587	215	13	v	v	ADP
ejpam-5587	215	14	such	such	ADJ
ejpam-5587	215	15	that	that	SCONJ
ejpam-5587	215	16	(	(	PUNCT
ejpam-5587	215	17	t11	t11	NOUN
ejpam-5587	215	18	−	−	NOUN
ejpam-5587	215	19	λi)∗u	λi)∗u	NOUN
ejpam-5587	215	20	=	=	PUNCT
ejpam-5587	215	21	(	(	PUNCT
ejpam-5587	215	22	t11	t11	NOUN
ejpam-5587	215	23	−	−	PROPN
ejpam-5587	215	24	λi)v	λi)v	PROPN
ejpam-5587	215	25	.	.	PUNCT
ejpam-5587	215	26	therefore	therefore	ADV
ejpam-5587	215	27	,	,	PUNCT
ejpam-5587	215	28	t12(y	t12(y	NUM
ejpam-5587	215	29	)	)	PUNCT
ejpam-5587	215	30	=	=	PUNCT
ejpam-5587	216	1	(	(	PUNCT
ejpam-5587	216	2	t11	t11	NOUN
ejpam-5587	216	3	−	−	PROPN
ejpam-5587	216	4	λi)v	λi)v	PROPN
ejpam-5587	216	5	for	for	ADP
ejpam-5587	216	6	every	every	DET
ejpam-5587	216	7	λ	λ	PROPN
ejpam-5587	216	8	∈	∈	PROPN
ejpam-5587	216	9	c.	c.	NOUN
ejpam-5587	216	10	consequently	consequently	ADV
ejpam-5587	216	11	,	,	PUNCT
ejpam-5587	216	12	t12(y	t12(y	PROPN
ejpam-5587	216	13	)	)	PUNCT
ejpam-5587	216	14	∈	∈	PROPN
ejpam-5587	217	1	⋂	⋂	PROPN
ejpam-5587	217	2	λ∈c	λ∈c	NOUN
ejpam-5587	217	3	ran(t11	ran(t11	NOUN
ejpam-5587	217	4	−	−	PROPN
ejpam-5587	217	5	λi	λi	NOUN
ejpam-5587	217	6	)	)	PUNCT
ejpam-5587	217	7	.	.	PUNCT
ejpam-5587	218	1	hence	hence	ADV
ejpam-5587	218	2	,	,	PUNCT
ejpam-5587	218	3	t12(y	t12(y	PRON
ejpam-5587	218	4	)	)	PUNCT
ejpam-5587	218	5	=	=	SYM
ejpam-5587	218	6	0	0	NUM
ejpam-5587	219	1	for	for	ADP
ejpam-5587	219	2	all	all	DET
ejpam-5587	219	3	y	y	PROPN
ejpam-5587	219	4	∈	∈	PROPN
ejpam-5587	219	5	d(t	d(t	PROPN
ejpam-5587	219	6	)	)	PUNCT
ejpam-5587	219	7	∩m⊥	∩m⊥	VERB
ejpam-5587	219	8	(	(	PUNCT
ejpam-5587	219	9	see	see	VERB
ejpam-5587	219	10	[	[	X
ejpam-5587	219	11	19	19	NUM
ejpam-5587	219	12	]	]	PUNCT
ejpam-5587	219	13	)	)	PUNCT
ejpam-5587	219	14	and	and	CCONJ
ejpam-5587	219	15	so	so	ADV
ejpam-5587	219	16	t12	t12	PROPN
ejpam-5587	219	17	=	=	SYM
ejpam-5587	219	18	0	0	X
ejpam-5587	219	19	.	.	PUNCT
ejpam-5587	220	1	this	this	PRON
ejpam-5587	220	2	ends	end	VERB
ejpam-5587	220	3	the	the	DET
ejpam-5587	220	4	proof	proof	NOUN
ejpam-5587	220	5	.	.	PUNCT
ejpam-5587	221	1	the	the	DET
ejpam-5587	221	2	ascent	ascent	PROPN
ejpam-5587	221	3	p(t	p(t	PROPN
ejpam-5587	221	4	)	)	PUNCT
ejpam-5587	221	5	and	and	CCONJ
ejpam-5587	221	6	descent	descent	NOUN
ejpam-5587	221	7	q(t	q(t	PROPN
ejpam-5587	221	8	)	)	PUNCT
ejpam-5587	221	9	of	of	ADP
ejpam-5587	221	10	an	an	DET
ejpam-5587	221	11	operator	operator	NOUN
ejpam-5587	221	12	c(h	c(h	NOUN
ejpam-5587	221	13	)	)	PUNCT
ejpam-5587	221	14	are	be	AUX
ejpam-5587	221	15	given	give	VERB
ejpam-5587	221	16	by	by	ADP
ejpam-5587	221	17	p(t	p(t	NOUN
ejpam-5587	221	18	)	)	PUNCT
ejpam-5587	222	1	=	=	SYM
ejpam-5587	222	2	inf	inf	NOUN
ejpam-5587	222	3	{	{	PUNCT
ejpam-5587	222	4	n	n	PROPN
ejpam-5587	222	5	:	:	PUNCT
ejpam-5587	222	6	ker(tn	ker(tn	X
ejpam-5587	222	7	)	)	PUNCT
ejpam-5587	222	8	=	=	SYM
ejpam-5587	222	9	ker(tn+1	ker(tn+1	PROPN
ejpam-5587	222	10	)	)	PUNCT
ejpam-5587	222	11	}	}	PUNCT
ejpam-5587	222	12	and	and	CCONJ
ejpam-5587	222	13	q(t	q(t	ADJ
ejpam-5587	222	14	)	)	PUNCT
ejpam-5587	222	15	=	=	SYM
ejpam-5587	222	16	inf	inf	NOUN
ejpam-5587	222	17	{	{	PUNCT
ejpam-5587	222	18	n	n	NOUN
ejpam-5587	222	19	:	:	PUNCT
ejpam-5587	222	20	ran(tn	ran(tn	NOUN
ejpam-5587	222	21	)	)	PUNCT
ejpam-5587	222	22	=	=	SYM
ejpam-5587	222	23	ran(tn+1	ran(tn+1	NOUN
ejpam-5587	222	24	)	)	PUNCT
ejpam-5587	222	25	}	}	PUNCT
ejpam-5587	223	1	it	it	PRON
ejpam-5587	223	2	follows	follow	VERB
ejpam-5587	223	3	from	from	ADP
ejpam-5587	223	4	the	the	DET
ejpam-5587	223	5	definition	definition	NOUN
ejpam-5587	223	6	of	of	ADP
ejpam-5587	223	7	totally	totally	ADV
ejpam-5587	223	8	paranormal	paranormal	ADJ
ejpam-5587	223	9	the	the	DET
ejpam-5587	223	10	following	follow	VERB
ejpam-5587	223	11	result	result	NOUN
ejpam-5587	223	12	holds	hold	VERB
ejpam-5587	223	13	.	.	PUNCT
ejpam-5587	224	1	s.	s.	PROPN
ejpam-5587	224	2	alnabulsi	alnabulsi	PROPN
ejpam-5587	224	3	,	,	PUNCT
ejpam-5587	224	4	m.h.m	m.h.m	PROPN
ejpam-5587	224	5	.	.	PUNCT
ejpam-5587	224	6	rashid	rashid	PROPN
ejpam-5587	224	7	/	/	SYM
ejpam-5587	224	8	eur	eur	PROPN
ejpam-5587	224	9	.	.	PUNCT
ejpam-5587	225	1	j.	j.	PROPN
ejpam-5587	225	2	pure	pure	PROPN
ejpam-5587	225	3	appl	appl	PROPN
ejpam-5587	225	4	.	.	PROPN
ejpam-5587	225	5	math	math	PROPN
ejpam-5587	225	6	,	,	PUNCT
ejpam-5587	225	7	18	18	NUM
ejpam-5587	225	8	(	(	PUNCT
ejpam-5587	225	9	1	1	NUM
ejpam-5587	225	10	)	)	PUNCT
ejpam-5587	225	11	(	(	PUNCT
ejpam-5587	225	12	2025	2025	NUM
ejpam-5587	225	13	)	)	PUNCT
ejpam-5587	225	14	,	,	PUNCT
ejpam-5587	225	15	5587	5587	NUM
ejpam-5587	225	16	8	8	NUM
ejpam-5587	225	17	of	of	ADP
ejpam-5587	225	18	20	20	NUM
ejpam-5587	225	19	proposition	proposition	NOUN
ejpam-5587	225	20	3	3	NUM
ejpam-5587	225	21	.	.	PUNCT
ejpam-5587	226	1	let	let	VERB
ejpam-5587	226	2	t	t	PROPN
ejpam-5587	226	3	∈	∈	PROPN
ejpam-5587	226	4	l(h	l(h	PROPN
ejpam-5587	226	5	)	)	PUNCT
ejpam-5587	226	6	be	be	AUX
ejpam-5587	226	7	a	a	DET
ejpam-5587	226	8	densely	densely	ADV
ejpam-5587	226	9	defined	define	VERB
ejpam-5587	226	10	closed	close	VERB
ejpam-5587	226	11	totally	totally	ADV
ejpam-5587	226	12	paranormal	paranormal	ADJ
ejpam-5587	226	13	operator	operator	NOUN
ejpam-5587	226	14	.	.	PUNCT
ejpam-5587	227	1	then	then	ADV
ejpam-5587	227	2	the	the	DET
ejpam-5587	227	3	ascent	ascent	PROPN
ejpam-5587	227	4	p(t	p(t	PROPN
ejpam-5587	227	5	)	)	PUNCT
ejpam-5587	227	6	and	and	CCONJ
ejpam-5587	227	7	descent	descent	NOUN
ejpam-5587	227	8	q(t	q(t	PROPN
ejpam-5587	227	9	)	)	PUNCT
ejpam-5587	227	10	of	of	ADP
ejpam-5587	227	11	t	t	PROPN
ejpam-5587	227	12	are	be	AUX
ejpam-5587	227	13	finite	finite	ADJ
ejpam-5587	227	14	for	for	ADP
ejpam-5587	227	15	all	all	DET
ejpam-5587	227	16	λ	λ	PROPN
ejpam-5587	227	17	∈	∈	PROPN
ejpam-5587	227	18	c.	c.	NOUN
ejpam-5587	227	19	definition	definition	NOUN
ejpam-5587	227	20	7	7	NUM
ejpam-5587	227	21	.	.	PUNCT
ejpam-5587	228	1	[	[	X
ejpam-5587	228	2	14	14	NUM
ejpam-5587	228	3	]	]	PUNCT
ejpam-5587	228	4	let	let	VERB
ejpam-5587	228	5	t	t	NOUN
ejpam-5587	228	6	be	be	AUX
ejpam-5587	228	7	a	a	DET
ejpam-5587	228	8	non	non	X
ejpam-5587	228	9	necessarily	necessarily	ADV
ejpam-5587	228	10	bounded	bound	VERB
ejpam-5587	228	11	operator	operator	NOUN
ejpam-5587	228	12	with	with	ADP
ejpam-5587	228	13	domain	domain	NOUN
ejpam-5587	228	14	d(t	d(t	PROPN
ejpam-5587	228	15	)	)	PUNCT
ejpam-5587	229	1	⊂	⊂	PROPN
ejpam-5587	230	1	h.	h.	PROPN
ejpam-5587	230	2	we	we	PRON
ejpam-5587	230	3	say	say	VERB
ejpam-5587	230	4	that	that	SCONJ
ejpam-5587	230	5	λ	λ	PROPN
ejpam-5587	230	6	is	be	AUX
ejpam-5587	230	7	not	not	PART
ejpam-5587	230	8	in	in	ADP
ejpam-5587	230	9	σ(t	σ(t	NOUN
ejpam-5587	230	10	)	)	PUNCT
ejpam-5587	231	1	if	if	SCONJ
ejpam-5587	231	2	t	t	PROPN
ejpam-5587	231	3	−	−	PROPN
ejpam-5587	231	4	λ	λ	PROPN
ejpam-5587	231	5	is	be	AUX
ejpam-5587	231	6	injective	injective	ADJ
ejpam-5587	231	7	and	and	CCONJ
ejpam-5587	231	8	(	(	PUNCT
ejpam-5587	231	9	t	t	PROPN
ejpam-5587	231	10	−	−	PROPN
ejpam-5587	231	11	λ)−1	λ)−1	NOUN
ejpam-5587	231	12	∈	∈	PROPN
ejpam-5587	231	13	b(h	b(h	PROPN
ejpam-5587	231	14	)	)	PUNCT
ejpam-5587	231	15	.	.	PUNCT
ejpam-5587	232	1	the	the	DET
ejpam-5587	232	2	following	follow	VERB
ejpam-5587	232	3	results	result	NOUN
ejpam-5587	232	4	immediately	immediately	ADV
ejpam-5587	232	5	follows	follow	VERB
ejpam-5587	232	6	from	from	ADP
ejpam-5587	232	7	the	the	DET
ejpam-5587	232	8	definition	definition	NOUN
ejpam-5587	232	9	.	.	PUNCT
ejpam-5587	233	1	proposition	proposition	NOUN
ejpam-5587	233	2	4	4	NUM
ejpam-5587	233	3	.	.	PUNCT
ejpam-5587	234	1	let	let	VERB
ejpam-5587	234	2	t	t	PROPN
ejpam-5587	234	3	∈	∈	PROPN
ejpam-5587	234	4	l(h	l(h	PROPN
ejpam-5587	234	5	)	)	PUNCT
ejpam-5587	234	6	be	be	AUX
ejpam-5587	234	7	a	a	DET
ejpam-5587	234	8	densely	densely	ADV
ejpam-5587	234	9	defined	define	VERB
ejpam-5587	234	10	closed	close	VERB
ejpam-5587	234	11	totally	totally	ADV
ejpam-5587	234	12	paranormal	paranormal	ADJ
ejpam-5587	234	13	operator	operator	NOUN
ejpam-5587	234	14	.	.	PUNCT
ejpam-5587	235	1	then	then	ADV
ejpam-5587	235	2	t	t	PROPN
ejpam-5587	235	3	−	−	PROPN
ejpam-5587	235	4	αi	αi	VERB
ejpam-5587	235	5	and	and	CCONJ
ejpam-5587	235	6	αt	αt	NOUN
ejpam-5587	235	7	are	be	AUX
ejpam-5587	235	8	totally	totally	ADV
ejpam-5587	235	9	paranormal	paranormal	ADJ
ejpam-5587	235	10	operators	operator	NOUN
ejpam-5587	235	11	for	for	ADP
ejpam-5587	235	12	all	all	DET
ejpam-5587	235	13	α	α	PRON
ejpam-5587	235	14	∈	∈	PROPN
ejpam-5587	235	15	c.	c.	NOUN
ejpam-5587	235	16	proposition	proposition	NOUN
ejpam-5587	235	17	5	5	NUM
ejpam-5587	235	18	.	.	PUNCT
ejpam-5587	236	1	let	let	AUX
ejpam-5587	236	2	let	let	VERB
ejpam-5587	236	3	t	t	PROPN
ejpam-5587	236	4	∈	∈	PROPN
ejpam-5587	236	5	l(h	l(h	PROPN
ejpam-5587	236	6	)	)	PUNCT
ejpam-5587	236	7	be	be	AUX
ejpam-5587	236	8	a	a	DET
ejpam-5587	236	9	densely	densely	ADV
ejpam-5587	236	10	defined	define	VERB
ejpam-5587	236	11	closed	close	VERB
ejpam-5587	236	12	totally	totally	ADV
ejpam-5587	236	13	paranormal	paranormal	ADJ
ejpam-5587	236	14	operator	operator	NOUN
ejpam-5587	236	15	.	.	PUNCT
ejpam-5587	237	1	if	if	SCONJ
ejpam-5587	237	2	σ(t	σ(t	PROPN
ejpam-5587	237	3	)	)	PUNCT
ejpam-5587	238	1	=	=	PUNCT
ejpam-5587	238	2	{	{	PUNCT
ejpam-5587	238	3	µ	µ	NOUN
ejpam-5587	238	4	}	}	PUNCT
ejpam-5587	238	5	,	,	PUNCT
ejpam-5587	238	6	then	then	ADV
ejpam-5587	238	7	t	t	PROPN
ejpam-5587	238	8	=	=	PUNCT
ejpam-5587	239	1	µi	µi	PROPN
ejpam-5587	239	2	.	.	PUNCT
ejpam-5587	240	1	we	we	PRON
ejpam-5587	240	2	now	now	ADV
ejpam-5587	240	3	give	give	VERB
ejpam-5587	240	4	the	the	DET
ejpam-5587	240	5	counterexample	counterexample	NOUN
ejpam-5587	240	6	of	of	ADP
ejpam-5587	240	7	a	a	DET
ejpam-5587	240	8	closed	close	VERB
ejpam-5587	240	9	densely	densely	ADV
ejpam-5587	240	10	defined	define	VERB
ejpam-5587	240	11	operator	operator	NOUN
ejpam-5587	240	12	t	t	NOUN
ejpam-5587	240	13	such	such	ADJ
ejpam-5587	240	14	that	that	SCONJ
ejpam-5587	240	15	both	both	DET
ejpam-5587	240	16	t	t	PROPN
ejpam-5587	240	17	and	and	CCONJ
ejpam-5587	240	18	t	t	PROPN
ejpam-5587	240	19	∗	∗	NOUN
ejpam-5587	240	20	are	be	AUX
ejpam-5587	240	21	one	one	NUM
ejpam-5587	240	22	-	-	PUNCT
ejpam-5587	240	23	to	to	ADP
ejpam-5587	240	24	-	-	PUNCT
ejpam-5587	240	25	one	one	NUM
ejpam-5587	240	26	and	and	CCONJ
ejpam-5587	240	27	totally	totally	ADV
ejpam-5587	240	28	paranormal	paranormal	ADJ
ejpam-5587	240	29	,	,	PUNCT
ejpam-5587	240	30	yet	yet	CCONJ
ejpam-5587	240	31	t	t	PROPN
ejpam-5587	240	32	is	be	AUX
ejpam-5587	240	33	not	not	PART
ejpam-5587	240	34	normal	normal	ADJ
ejpam-5587	240	35	.	.	PUNCT
ejpam-5587	240	36	example	example	NOUN
ejpam-5587	241	1	1	1	NUM
ejpam-5587	241	2	.	.	PUNCT
ejpam-5587	242	1	the	the	DET
ejpam-5587	242	2	hilbert	hilbert	PROPN
ejpam-5587	242	3	space	space	NOUN
ejpam-5587	242	4	in	in	ADP
ejpam-5587	242	5	question	question	NOUN
ejpam-5587	242	6	is	be	AUX
ejpam-5587	242	7	l2(r)⊕l2(r	l2(r)⊕l2(r	PROPN
ejpam-5587	242	8	)	)	PUNCT
ejpam-5587	242	9	.	.	PUNCT
ejpam-5587	243	1	from	from	ADP
ejpam-5587	243	2	[	[	X
ejpam-5587	243	3	24	24	NUM
ejpam-5587	243	4	]	]	PUNCT
ejpam-5587	243	5	,	,	PUNCT
ejpam-5587	243	6	we	we	PRON
ejpam-5587	243	7	have	have	VERB
ejpam-5587	243	8	an	an	DET
ejpam-5587	243	9	explicit	explicit	ADJ
ejpam-5587	243	10	example	example	NOUN
ejpam-5587	243	11	of	of	ADP
ejpam-5587	243	12	a	a	DET
ejpam-5587	243	13	densely	densely	ADV
ejpam-5587	243	14	defined	define	VERB
ejpam-5587	243	15	unbounded	unbounded	ADJ
ejpam-5587	243	16	closed	closed	ADJ
ejpam-5587	243	17	operator	operator	NOUN
ejpam-5587	243	18	t	t	NOUN
ejpam-5587	243	19	for	for	ADP
ejpam-5587	243	20	which	which	PRON
ejpam-5587	243	21	d(t	d(t	PROPN
ejpam-5587	243	22	2	2	NUM
ejpam-5587	243	23	)	)	PUNCT
ejpam-5587	243	24	=	=	PROPN
ejpam-5587	243	25	d(t	d(t	PROPN
ejpam-5587	243	26	∗2	∗2	PROPN
ejpam-5587	243	27	)	)	PUNCT
ejpam-5587	243	28	=	=	PRON
ejpam-5587	244	1	{	{	PUNCT
ejpam-5587	244	2	0	0	NUM
ejpam-5587	244	3	}	}	PUNCT
ejpam-5587	244	4	more	more	ADV
ejpam-5587	244	5	precisely	precisely	ADV
ejpam-5587	244	6	,	,	PUNCT
ejpam-5587	244	7	t	t	PROPN
ejpam-5587	244	8	is	be	AUX
ejpam-5587	244	9	defined	define	VERB
ejpam-5587	244	10	by	by	ADP
ejpam-5587	244	11	t	t	X
ejpam-5587	244	12	=	=	PUNCT
ejpam-5587	245	1	[	[	PUNCT
ejpam-5587	245	2	0	0	NUM
ejpam-5587	245	3	a−1	a−1	PROPN
ejpam-5587	245	4	b	b	PROPN
ejpam-5587	245	5	0	0	NUM
ejpam-5587	245	6	]	]	PUNCT
ejpam-5587	245	7	on	on	ADP
ejpam-5587	245	8	d(t	d(t	PROPN
ejpam-5587	245	9	)	)	PUNCT
ejpam-5587	245	10	:	:	PUNCT
ejpam-5587	245	11	=	=	SYM
ejpam-5587	245	12	d(b	d(b	X
ejpam-5587	245	13	)	)	PUNCT
ejpam-5587	245	14	⊕	⊕	PROPN
ejpam-5587	245	15	d(a−1	d(a−1	PROPN
ejpam-5587	245	16	)	)	PUNCT
ejpam-5587	245	17	⊂	⊂	PROPN
ejpam-5587	246	1	l2(r	l2(r	X
ejpam-5587	246	2	)	)	PUNCT
ejpam-5587	246	3	⊕	⊕	PROPN
ejpam-5587	246	4	l2(r	l2(r	NOUN
ejpam-5587	246	5	)	)	PUNCT
ejpam-5587	246	6	,	,	PUNCT
ejpam-5587	246	7	and	and	CCONJ
ejpam-5587	246	8	where	where	SCONJ
ejpam-5587	246	9	a	a	PRON
ejpam-5587	246	10	and	and	CCONJ
ejpam-5587	246	11	b	b	NOUN
ejpam-5587	246	12	are	be	AUX
ejpam-5587	246	13	two	two	NUM
ejpam-5587	246	14	unbounded	unbounded	ADJ
ejpam-5587	246	15	self	self	NOUN
ejpam-5587	246	16	-	-	PUNCT
ejpam-5587	246	17	adjoint	adjoint	NOUN
ejpam-5587	246	18	operators	operator	NOUN
ejpam-5587	246	19	such	such	ADJ
ejpam-5587	246	20	that	that	SCONJ
ejpam-5587	246	21	d(a	d(a	PROPN
ejpam-5587	246	22	)	)	PUNCT
ejpam-5587	246	23	∩d(b	∩d(b	PROPN
ejpam-5587	246	24	)	)	PUNCT
ejpam-5587	246	25	=	=	SYM
ejpam-5587	246	26	d(a−1	d(a−1	NOUN
ejpam-5587	246	27	)	)	PUNCT
ejpam-5587	246	28	∩d(b−1	∩d(b−1	NOUN
ejpam-5587	246	29	)	)	PUNCT
ejpam-5587	247	1	=	=	PRON
ejpam-5587	247	2	{	{	PUNCT
ejpam-5587	247	3	0	0	NUM
ejpam-5587	247	4	}	}	PUNCT
ejpam-5587	247	5	,	,	PUNCT
ejpam-5587	247	6	where	where	SCONJ
ejpam-5587	247	7	a−1	a−1	PROPN
ejpam-5587	247	8	and	and	CCONJ
ejpam-5587	247	9	b−1	b−1	PROPN
ejpam-5587	247	10	are	be	AUX
ejpam-5587	247	11	not	not	PART
ejpam-5587	247	12	bounded	bound	VERB
ejpam-5587	247	13	(	(	PUNCT
ejpam-5587	247	14	as	as	ADP
ejpam-5587	247	15	in	in	ADP
ejpam-5587	247	16	[	[	X
ejpam-5587	247	17	15	15	NUM
ejpam-5587	247	18	]	]	NUM
ejpam-5587	247	19	)	)	PUNCT
ejpam-5587	247	20	.	.	PUNCT
ejpam-5587	248	1	hence	hence	ADV
ejpam-5587	248	2	t	t	NOUN
ejpam-5587	248	3	∗	∗	NOUN
ejpam-5587	248	4	=	=	PUNCT
ejpam-5587	249	1	[	[	PUNCT
ejpam-5587	249	2	0	0	NUM
ejpam-5587	249	3	b	b	X
ejpam-5587	249	4	a−1	a−1	PROPN
ejpam-5587	249	5	0	0	NUM
ejpam-5587	249	6	]	]	PUNCT
ejpam-5587	249	7	for	for	ADP
ejpam-5587	249	8	a−1	a−1	PROPN
ejpam-5587	249	9	and	and	CCONJ
ejpam-5587	249	10	b	b	PROPN
ejpam-5587	249	11	are	be	AUX
ejpam-5587	249	12	both	both	PRON
ejpam-5587	249	13	self	self	NOUN
ejpam-5587	249	14	-	-	PUNCT
ejpam-5587	249	15	adjoint	adjoint	NOUN
ejpam-5587	249	16	.	.	PUNCT
ejpam-5587	250	1	observe	observe	VERB
ejpam-5587	250	2	now	now	ADV
ejpam-5587	250	3	that	that	SCONJ
ejpam-5587	250	4	both	both	DET
ejpam-5587	250	5	t	t	PROPN
ejpam-5587	250	6	and	and	CCONJ
ejpam-5587	250	7	t	t	PROPN
ejpam-5587	250	8	∗	∗	NOUN
ejpam-5587	250	9	are	be	AUX
ejpam-5587	250	10	one	one	NUM
ejpam-5587	250	11	-	-	PUNCT
ejpam-5587	250	12	to	to	ADP
ejpam-5587	250	13	-	-	PUNCT
ejpam-5587	250	14	one	one	NOUN
ejpam-5587	250	15	since	since	SCONJ
ejpam-5587	250	16	both	both	CCONJ
ejpam-5587	250	17	a−1	a−1	PROPN
ejpam-5587	250	18	and	and	CCONJ
ejpam-5587	250	19	b	b	NOUN
ejpam-5587	250	20	are	be	AUX
ejpam-5587	250	21	so	so	ADV
ejpam-5587	250	22	.	.	PUNCT
ejpam-5587	251	1	both	both	DET
ejpam-5587	251	2	t	t	PROPN
ejpam-5587	251	3	and	and	CCONJ
ejpam-5587	251	4	t	t	PROPN
ejpam-5587	251	5	∗	∗	NOUN
ejpam-5587	251	6	are	be	AUX
ejpam-5587	251	7	trivially	trivially	ADV
ejpam-5587	251	8	totally	totally	ADV
ejpam-5587	251	9	paranormal	paranormal	ADJ
ejpam-5587	251	10	thanks	thank	NOUN
ejpam-5587	251	11	to	to	ADP
ejpam-5587	251	12	the	the	DET
ejpam-5587	251	13	assumption	assumption	NOUN
ejpam-5587	251	14	d(t	d(t	PROPN
ejpam-5587	251	15	2	2	NUM
ejpam-5587	251	16	)	)	PUNCT
ejpam-5587	251	17	=	=	PROPN
ejpam-5587	251	18	d(t	d(t	PROPN
ejpam-5587	251	19	∗2	∗2	PROPN
ejpam-5587	251	20	)	)	PUNCT
ejpam-5587	251	21	=	=	PRON
ejpam-5587	251	22	{	{	PUNCT
ejpam-5587	251	23	0	0	NUM
ejpam-5587	251	24	}	}	PUNCT
ejpam-5587	251	25	.	.	PUNCT
ejpam-5587	252	1	so	so	ADV
ejpam-5587	252	2	tatally	tatally	ADV
ejpam-5587	252	3	paranormality	paranormality	NOUN
ejpam-5587	252	4	of	of	ADP
ejpam-5587	252	5	both	both	DET
ejpam-5587	252	6	operators	operator	NOUN
ejpam-5587	252	7	need	need	AUX
ejpam-5587	252	8	only	only	ADV
ejpam-5587	252	9	be	be	AUX
ejpam-5587	252	10	checked	check	VERB
ejpam-5587	252	11	at	at	ADP
ejpam-5587	252	12	the	the	DET
ejpam-5587	252	13	zero	zero	NUM
ejpam-5587	252	14	vector	vector	NOUN
ejpam-5587	252	15	and	and	CCONJ
ejpam-5587	252	16	this	this	PRON
ejpam-5587	252	17	is	be	AUX
ejpam-5587	252	18	plain	plain	ADJ
ejpam-5587	252	19	as	as	ADP
ejpam-5587	252	20	∥(t	∥(t	NOUN
ejpam-5587	252	21	−	−	NOUN
ejpam-5587	252	22	µi)x∥2	µi)x∥2	NOUN
ejpam-5587	252	23	=	=	SYM
ejpam-5587	252	24	∥∥(t	∥∥(t	ADJ
ejpam-5587	252	25	−	−	NOUN
ejpam-5587	252	26	µi)2x	µi)2x	VERB
ejpam-5587	252	27	∥∥	∥∥	X
ejpam-5587	252	28	∥x∥	∥x∥	NOUN
ejpam-5587	252	29	=	=	SYM
ejpam-5587	252	30	0	0	PUNCT
ejpam-5587	252	31	and	and	CCONJ
ejpam-5587	252	32	∥(t	∥(t	VERB
ejpam-5587	252	33	−	−	X
ejpam-5587	252	34	µi)∗x∥2	µi)∗x∥2	NOUN
ejpam-5587	252	35	=	=	PUNCT
ejpam-5587	252	36	∥∥(t	∥∥(t	PRON
ejpam-5587	252	37	−	−	NOUN
ejpam-5587	252	38	µi)2x	µi)2x	VERB
ejpam-5587	252	39	∥∥	∥∥	X
ejpam-5587	252	40	∥x∥	∥x∥	NOUN
ejpam-5587	252	41	=	=	SYM
ejpam-5587	252	42	0	0	NUM
ejpam-5587	252	43	for	for	ADP
ejpam-5587	252	44	x	x	X
ejpam-5587	252	45	=	=	SYM
ejpam-5587	252	46	0	0	NUM
ejpam-5587	252	47	.	.	PUNCT
ejpam-5587	253	1	however	however	ADV
ejpam-5587	253	2	,	,	PUNCT
ejpam-5587	253	3	t	t	PROPN
ejpam-5587	253	4	can	can	AUX
ejpam-5587	253	5	not	not	PART
ejpam-5587	253	6	be	be	AUX
ejpam-5587	253	7	normal	normal	ADJ
ejpam-5587	253	8	for	for	SCONJ
ejpam-5587	253	9	it	it	PRON
ejpam-5587	253	10	were	be	AUX
ejpam-5587	253	11	,	,	PUNCT
ejpam-5587	253	12	t	t	PROPN
ejpam-5587	253	13	2	2	NUM
ejpam-5587	253	14	would	would	AUX
ejpam-5587	253	15	be	be	AUX
ejpam-5587	253	16	normal	normal	ADJ
ejpam-5587	253	17	too	too	ADV
ejpam-5587	253	18	,	,	PUNCT
ejpam-5587	253	19	in	in	ADP
ejpam-5587	253	20	particular	particular	ADJ
ejpam-5587	253	21	it	it	PRON
ejpam-5587	253	22	would	would	AUX
ejpam-5587	253	23	be	be	AUX
ejpam-5587	253	24	densely	densely	ADV
ejpam-5587	253	25	defined	define	VERB
ejpam-5587	253	26	which	which	PRON
ejpam-5587	253	27	is	be	AUX
ejpam-5587	253	28	impossible	impossible	ADJ
ejpam-5587	253	29	here	here	ADV
ejpam-5587	253	30	here	here	ADV
ejpam-5587	253	31	we	we	PRON
ejpam-5587	253	32	discuss	discuss	VERB
ejpam-5587	253	33	some	some	DET
ejpam-5587	253	34	basic	basic	ADJ
ejpam-5587	253	35	results	result	NOUN
ejpam-5587	253	36	related	relate	VERB
ejpam-5587	253	37	to	to	ADP
ejpam-5587	253	38	unbounded	unbounded	ADJ
ejpam-5587	253	39	totally	totally	ADV
ejpam-5587	253	40	paranormal	paranormal	ADJ
ejpam-5587	253	41	operators	operator	NOUN
ejpam-5587	253	42	,	,	PUNCT
ejpam-5587	253	43	which	which	PRON
ejpam-5587	253	44	are	be	AUX
ejpam-5587	253	45	often	often	ADV
ejpam-5587	253	46	used	use	VERB
ejpam-5587	253	47	in	in	ADP
ejpam-5587	253	48	the	the	DET
ejpam-5587	253	49	article	article	NOUN
ejpam-5587	253	50	.	.	PUNCT
ejpam-5587	254	1	theorem	theorem	PROPN
ejpam-5587	254	2	5	5	NUM
ejpam-5587	254	3	.	.	PUNCT
ejpam-5587	255	1	let	let	VERB
ejpam-5587	255	2	t	t	PROPN
ejpam-5587	255	3	∈	∈	PROPN
ejpam-5587	255	4	l(h	l(h	PROPN
ejpam-5587	255	5	)	)	PUNCT
ejpam-5587	255	6	be	be	AUX
ejpam-5587	255	7	a	a	DET
ejpam-5587	255	8	densely	densely	ADV
ejpam-5587	255	9	defined	define	VERB
ejpam-5587	255	10	closed	close	VERB
ejpam-5587	255	11	totally	totally	ADV
ejpam-5587	255	12	paranormal	paranormal	ADJ
ejpam-5587	255	13	operator	operator	NOUN
ejpam-5587	255	14	and	and	CCONJ
ejpam-5587	255	15	not	not	PART
ejpam-5587	255	16	a	a	DET
ejpam-5587	255	17	multiple	multiple	NOUN
ejpam-5587	255	18	of	of	ADP
ejpam-5587	255	19	the	the	DET
ejpam-5587	255	20	identity	identity	NOUN
ejpam-5587	255	21	.	.	PUNCT
ejpam-5587	256	1	then	then	ADV
ejpam-5587	256	2	the	the	DET
ejpam-5587	256	3	following	follow	VERB
ejpam-5587	256	4	holds	hold	VERB
ejpam-5587	256	5	.	.	PUNCT
ejpam-5587	257	1	(	(	PUNCT
ejpam-5587	257	2	i	i	NOUN
ejpam-5587	257	3	)	)	PUNCT
ejpam-5587	257	4	if	if	SCONJ
ejpam-5587	257	5	m	m	NOUN
ejpam-5587	257	6	is	be	AUX
ejpam-5587	257	7	a	a	DET
ejpam-5587	257	8	closed	close	VERB
ejpam-5587	257	9	invariant	invariant	ADJ
ejpam-5587	257	10	subspace	subspace	NOUN
ejpam-5587	257	11	of	of	ADP
ejpam-5587	257	12	t	t	PROPN
ejpam-5587	257	13	,	,	PUNCT
ejpam-5587	257	14	then	then	ADV
ejpam-5587	257	15	t	t	PROPN
ejpam-5587	257	16	|m	|m	NOUN
ejpam-5587	257	17	is	be	AUX
ejpam-5587	257	18	totally	totally	ADV
ejpam-5587	257	19	paranormal	paranormal	ADJ
ejpam-5587	257	20	.	.	PUNCT
ejpam-5587	258	1	(	(	PUNCT
ejpam-5587	258	2	ii	ii	NOUN
ejpam-5587	258	3	)	)	PUNCT
ejpam-5587	258	4	if	if	SCONJ
ejpam-5587	258	5	0	0	NUM
ejpam-5587	258	6	/∈	/∈	PUNCT
ejpam-5587	258	7	σ(t	σ(t	PROPN
ejpam-5587	258	8	)	)	PUNCT
ejpam-5587	258	9	,	,	PUNCT
ejpam-5587	258	10	then	then	ADV
ejpam-5587	258	11	t−1	t−1	PROPN
ejpam-5587	258	12	totally	totally	ADV
ejpam-5587	258	13	paranormal	paranormal	PROPN
ejpam-5587	258	14	s.	s.	PROPN
ejpam-5587	258	15	alnabulsi	alnabulsi	PROPN
ejpam-5587	258	16	,	,	PUNCT
ejpam-5587	258	17	m.h.m	m.h.m	PROPN
ejpam-5587	258	18	.	.	PUNCT
ejpam-5587	258	19	rashid	rashid	PROPN
ejpam-5587	258	20	/	/	SYM
ejpam-5587	258	21	eur	eur	PROPN
ejpam-5587	258	22	.	.	PUNCT
ejpam-5587	259	1	j.	j.	PROPN
ejpam-5587	259	2	pure	pure	PROPN
ejpam-5587	259	3	appl	appl	PROPN
ejpam-5587	259	4	.	.	PROPN
ejpam-5587	259	5	math	math	PROPN
ejpam-5587	259	6	,	,	PUNCT
ejpam-5587	259	7	18	18	NUM
ejpam-5587	259	8	(	(	PUNCT
ejpam-5587	259	9	1	1	NUM
ejpam-5587	259	10	)	)	PUNCT
ejpam-5587	259	11	(	(	PUNCT
ejpam-5587	259	12	2025	2025	NUM
ejpam-5587	259	13	)	)	PUNCT
ejpam-5587	259	14	,	,	PUNCT
ejpam-5587	259	15	5587	5587	NUM
ejpam-5587	259	16	9	9	NUM
ejpam-5587	259	17	of	of	ADP
ejpam-5587	259	18	20	20	NUM
ejpam-5587	259	19	(	(	PUNCT
ejpam-5587	259	20	iii	iii	NOUN
ejpam-5587	259	21	)	)	PUNCT
ejpam-5587	259	22	σ(t	σ(t	PROPN
ejpam-5587	259	23	)	)	PUNCT
ejpam-5587	259	24	is	be	AUX
ejpam-5587	259	25	nonempty	nonempty	ADJ
ejpam-5587	259	26	.	.	PUNCT
ejpam-5587	260	1	proof	proof	NOUN
ejpam-5587	260	2	.	.	PUNCT
ejpam-5587	261	1	(	(	PUNCT
ejpam-5587	261	2	i	i	NOUN
ejpam-5587	261	3	)	)	PUNCT
ejpam-5587	261	4	for	for	ADP
ejpam-5587	261	5	every	every	DET
ejpam-5587	261	6	λ	λ	PROPN
ejpam-5587	261	7	∈	∈	PROPN
ejpam-5587	261	8	c	c	NOUN
ejpam-5587	261	9	,	,	PUNCT
ejpam-5587	261	10	as	as	SCONJ
ejpam-5587	261	11	m	m	PROPN
ejpam-5587	261	12	is	be	AUX
ejpam-5587	261	13	invariant	invariant	ADJ
ejpam-5587	261	14	under	under	ADP
ejpam-5587	261	15	t	t	PROPN
ejpam-5587	261	16	,	,	PUNCT
ejpam-5587	261	17	we	we	PRON
ejpam-5587	261	18	have	have	AUX
ejpam-5587	261	19	d((t	d((t	VERB
ejpam-5587	261	20	−	−	NOUN
ejpam-5587	261	21	λi)2|m	λi)2|m	ADJ
ejpam-5587	261	22	)	)	PUNCT
ejpam-5587	262	1	=	=	SYM
ejpam-5587	262	2	d	d	PROPN
ejpam-5587	262	3	(	(	PUNCT
ejpam-5587	262	4	(	(	PUNCT
ejpam-5587	262	5	t	t	NOUN
ejpam-5587	262	6	−	−	NOUN
ejpam-5587	262	7	λi)2	λi)2	PROPN
ejpam-5587	262	8	)	)	PUNCT
ejpam-5587	262	9	∩m	∩m	PROPN
ejpam-5587	263	1	=	=	PUNCT
ejpam-5587	263	2	{	{	PUNCT
ejpam-5587	263	3	x	x	PROPN
ejpam-5587	263	4	∈	∈	PROPN
ejpam-5587	263	5	d(t	d(t	PROPN
ejpam-5587	263	6	−	−	PROPN
ejpam-5587	263	7	λi	λi	NOUN
ejpam-5587	263	8	)	)	PUNCT
ejpam-5587	263	9	:	:	PUNCT
ejpam-5587	263	10	(	(	PUNCT
ejpam-5587	263	11	t	t	X
ejpam-5587	263	12	−	−	PROPN
ejpam-5587	263	13	λi)x	λi)x	PROPN
ejpam-5587	263	14	∈	∈	PROPN
ejpam-5587	263	15	d(t	d(t	PROPN
ejpam-5587	263	16	−	−	PROPN
ejpam-5587	263	17	λi	λi	NOUN
ejpam-5587	263	18	)	)	PUNCT
ejpam-5587	263	19	}	}	PUNCT
ejpam-5587	263	20	∩m	∩m	PROPN
ejpam-5587	264	1	=	=	PRON
ejpam-5587	264	2	{	{	PUNCT
ejpam-5587	264	3	x	x	PROPN
ejpam-5587	264	4	∈	∈	PROPN
ejpam-5587	264	5	d(t	d(t	PROPN
ejpam-5587	264	6	−	−	PROPN
ejpam-5587	264	7	λi	λi	NOUN
ejpam-5587	264	8	)	)	PUNCT
ejpam-5587	264	9	∩m	∩m	NOUN
ejpam-5587	264	10	:	:	PUNCT
ejpam-5587	264	11	(	(	PUNCT
ejpam-5587	264	12	t	t	X
ejpam-5587	264	13	−	−	PROPN
ejpam-5587	264	14	λi)x	λi)x	PROPN
ejpam-5587	264	15	∈	∈	PROPN
ejpam-5587	264	16	d(t	d(t	PROPN
ejpam-5587	264	17	−	−	PROPN
ejpam-5587	264	18	λi	λi	NOUN
ejpam-5587	264	19	)	)	PUNCT
ejpam-5587	264	20	∩m	∩m	PROPN
ejpam-5587	264	21	}	}	PUNCT
ejpam-5587	264	22	(	(	PUNCT
ejpam-5587	264	23	since	since	SCONJ
ejpam-5587	264	24	t	t	PROPN
ejpam-5587	264	25	(	(	PUNCT
ejpam-5587	264	26	d(t	d(t	PROPN
ejpam-5587	264	27	−	−	PROPN
ejpam-5587	264	28	λi	λi	NOUN
ejpam-5587	264	29	)	)	PUNCT
ejpam-5587	264	30	∩m	∩m	PROPN
ejpam-5587	264	31	)	)	PUNCT
ejpam-5587	265	1	⊆	⊆	NUM
ejpam-5587	265	2	m	m	NOUN
ejpam-5587	265	3	)	)	PUNCT
ejpam-5587	266	1	=	=	PRON
ejpam-5587	266	2	{	{	PUNCT
ejpam-5587	266	3	x	x	PUNCT
ejpam-5587	266	4	∈	∈	PROPN
ejpam-5587	266	5	d((t	d((t	NOUN
ejpam-5587	266	6	−	−	NOUN
ejpam-5587	266	7	λi)|m	λi)|m	NOUN
ejpam-5587	266	8	)	)	PUNCT
ejpam-5587	266	9	:	:	PUNCT
ejpam-5587	266	10	tx	tx	PROPN
ejpam-5587	266	11	∈	∈	PROPN
ejpam-5587	266	12	d((t	d((t	NOUN
ejpam-5587	266	13	−	−	NOUN
ejpam-5587	266	14	λi)|m	λi)|m	NOUN
ejpam-5587	266	15	)	)	PUNCT
ejpam-5587	266	16	}	}	PUNCT
ejpam-5587	267	1	=	=	SYM
ejpam-5587	267	2	d	d	NOUN
ejpam-5587	267	3	(	(	PUNCT
ejpam-5587	267	4	(	(	PUNCT
ejpam-5587	267	5	(	(	PUNCT
ejpam-5587	267	6	t	t	NOUN
ejpam-5587	267	7	−	−	PROPN
ejpam-5587	267	8	λi)|m)2	λi)|m)2	PROPN
ejpam-5587	267	9	)	)	PUNCT
ejpam-5587	267	10	.	.	PUNCT
ejpam-5587	268	1	thus	thus	ADV
ejpam-5587	268	2	,	,	PUNCT
ejpam-5587	268	3	(	(	PUNCT
ejpam-5587	268	4	t	t	NOUN
ejpam-5587	268	5	−	−	NOUN
ejpam-5587	268	6	λi)2|m	λi)2|m	NOUN
ejpam-5587	268	7	=	=	X
ejpam-5587	268	8	(	(	PUNCT
ejpam-5587	268	9	(	(	PUNCT
ejpam-5587	268	10	t	t	PROPN
ejpam-5587	268	11	−	−	PROPN
ejpam-5587	268	12	λ)|m)2	λ)|m)2	PROPN
ejpam-5587	268	13	.	.	PUNCT
ejpam-5587	269	1	now	now	ADV
ejpam-5587	269	2	the	the	DET
ejpam-5587	269	3	result	result	NOUN
ejpam-5587	269	4	follows	follow	VERB
ejpam-5587	269	5	from	from	ADP
ejpam-5587	269	6	the	the	DET
ejpam-5587	269	7	below	below	ADJ
ejpam-5587	269	8	inequality	inequality	NOUN
ejpam-5587	269	9	;	;	PUNCT
ejpam-5587	269	10	∥(t	∥(t	VERB
ejpam-5587	269	11	−	−	NOUN
ejpam-5587	269	12	λi)|mx∥2	λi)|mx∥2	NOUN
ejpam-5587	269	13	=	=	SYM
ejpam-5587	269	14	∥(t	∥(t	NOUN
ejpam-5587	270	1	−	−	X
ejpam-5587	270	2	λi)x∥2	λi)x∥2	NOUN
ejpam-5587	270	3	≤	≤	ADJ
ejpam-5587	270	4	∥∥(t	∥∥(t	PUNCT
ejpam-5587	270	5	−	−	NOUN
ejpam-5587	270	6	λi)2x	λi)2x	X
ejpam-5587	270	7	∥∥	∥∥	NOUN
ejpam-5587	270	8	=	=	SYM
ejpam-5587	270	9	∥∥(t	∥∥(t	ADJ
ejpam-5587	270	10	−	−	NOUN
ejpam-5587	270	11	λi)2|mx	λi)2|mx	PROPN
ejpam-5587	270	12	∥∥	∥∥	X
ejpam-5587	270	13	=	=	NOUN
ejpam-5587	270	14	∥∥∥((t	∥∥∥((t	NOUN
ejpam-5587	271	1	−	−	PROPN
ejpam-5587	272	1	λi)|m)2	λi)|m)2	PROPN
ejpam-5587	272	2	x	x	SYM
ejpam-5587	272	3	∥∥∥	∥∥∥	PROPN
ejpam-5587	272	4	,	,	PUNCT
ejpam-5587	272	5	∀x	∀x	X
ejpam-5587	272	6	∈	∈	PROPN
ejpam-5587	272	7	td(((t−λi)|m)2	td(((t−λi)|m)2	NOUN
ejpam-5587	272	8	)	)	PUNCT
ejpam-5587	272	9	.	.	PUNCT
ejpam-5587	273	1	(	(	PUNCT
ejpam-5587	273	2	ii	ii	NOUN
ejpam-5587	273	3	)	)	PUNCT
ejpam-5587	273	4	existence	existence	NOUN
ejpam-5587	273	5	of	of	ADP
ejpam-5587	273	6	t−1	t−1	PROPN
ejpam-5587	273	7	implies	imply	VERB
ejpam-5587	273	8	ran(t	ran(t	X
ejpam-5587	273	9	)	)	PUNCT
ejpam-5587	274	1	=	=	SYM
ejpam-5587	274	2	h	h	NOUN
ejpam-5587	274	3	and	and	CCONJ
ejpam-5587	274	4	consequently	consequently	ADV
ejpam-5587	274	5	ran	run	VERB
ejpam-5587	274	6	(	(	PUNCT
ejpam-5587	274	7	(	(	PUNCT
ejpam-5587	274	8	t	t	NOUN
ejpam-5587	274	9	)	)	PUNCT
ejpam-5587	274	10	2	2	NUM
ejpam-5587	274	11	)	)	PUNCT
ejpam-5587	274	12	=	=	SYM
ejpam-5587	275	1	h.	h.	PROPN
ejpam-5587	275	2	as	as	SCONJ
ejpam-5587	275	3	t	t	PROPN
ejpam-5587	275	4	is	be	AUX
ejpam-5587	275	5	totally	totally	ADV
ejpam-5587	275	6	paranormal	paranormal	ADJ
ejpam-5587	275	7	,	,	PUNCT
ejpam-5587	275	8	we	we	PRON
ejpam-5587	275	9	get	get	VERB
ejpam-5587	275	10	ker(t	ker(t	NOUN
ejpam-5587	275	11	)	)	PUNCT
ejpam-5587	276	1	=	=	SYM
ejpam-5587	276	2	ker((t	ker((t	VERB
ejpam-5587	276	3	)	)	PUNCT
ejpam-5587	276	4	2	2	NUM
ejpam-5587	276	5	)	)	PUNCT
ejpam-5587	276	6	,	,	PUNCT
ejpam-5587	276	7	so	so	SCONJ
ejpam-5587	276	8	t	t	PROPN
ejpam-5587	276	9	2	2	NUM
ejpam-5587	276	10	is	be	AUX
ejpam-5587	276	11	bijective	bijective	ADJ
ejpam-5587	276	12	and	and	CCONJ
ejpam-5587	276	13	(	(	PUNCT
ejpam-5587	276	14	(	(	PUNCT
ejpam-5587	276	15	t	t	NOUN
ejpam-5587	276	16	)	)	PUNCT
ejpam-5587	276	17	2	2	NUM
ejpam-5587	276	18	)	)	PUNCT
ejpam-5587	276	19	−1	−1	NOUN
ejpam-5587	276	20	exists	exist	VERB
ejpam-5587	276	21	.	.	PUNCT
ejpam-5587	277	1	also	also	ADV
ejpam-5587	277	2	d	d	X
ejpam-5587	277	3	(	(	PUNCT
ejpam-5587	277	4	(	(	PUNCT
ejpam-5587	277	5	(	(	PUNCT
ejpam-5587	277	6	t	t	NOUN
ejpam-5587	277	7	)	)	PUNCT
ejpam-5587	277	8	−1	−1	NOUN
ejpam-5587	277	9	)	)	PUNCT
ejpam-5587	277	10	2	2	X
ejpam-5587	277	11	)	)	PUNCT
ejpam-5587	277	12	=	=	SYM
ejpam-5587	277	13	h	h	NOUN
ejpam-5587	277	14	=	=	PRON
ejpam-5587	277	15	ran	run	VERB
ejpam-5587	277	16	(	(	PUNCT
ejpam-5587	277	17	(	(	PUNCT
ejpam-5587	277	18	t	t	NOUN
ejpam-5587	277	19	)	)	PUNCT
ejpam-5587	277	20	2	2	NUM
ejpam-5587	277	21	)	)	PUNCT
ejpam-5587	277	22	.	.	PUNCT
ejpam-5587	278	1	if	if	SCONJ
ejpam-5587	278	2	y	y	PROPN
ejpam-5587	278	3	∈	∈	PROPN
ejpam-5587	278	4	h	h	NOUN
ejpam-5587	278	5	,	,	PUNCT
ejpam-5587	278	6	then	then	ADV
ejpam-5587	278	7	there	there	PRON
ejpam-5587	278	8	exist	exist	VERB
ejpam-5587	278	9	x	x	PUNCT
ejpam-5587	278	10	∈	∈	PROPN
ejpam-5587	278	11	d	d	X
ejpam-5587	278	12	(	(	PUNCT
ejpam-5587	278	13	(	(	PUNCT
ejpam-5587	278	14	t	t	NOUN
ejpam-5587	278	15	)	)	PUNCT
ejpam-5587	278	16	2	2	NUM
ejpam-5587	278	17	)	)	PUNCT
ejpam-5587	278	18	,	,	PUNCT
ejpam-5587	278	19	such	such	ADJ
ejpam-5587	278	20	that	that	SCONJ
ejpam-5587	278	21	y	y	PROPN
ejpam-5587	278	22	=	=	SYM
ejpam-5587	278	23	t	t	PROPN
ejpam-5587	278	24	2x	2x	NUM
ejpam-5587	278	25	.	.	PUNCT
ejpam-5587	279	1	now	now	ADV
ejpam-5587	279	2	,	,	PUNCT
ejpam-5587	279	3	∥∥t−1y	∥∥t−1y	PROPN
ejpam-5587	279	4	∥∥2	∥∥2	PROPN
ejpam-5587	279	5	=	=	SYM
ejpam-5587	279	6	∥tx∥2	∥tx∥2	PROPN
ejpam-5587	279	7	≤	≤	NUM
ejpam-5587	279	8	∥∥t	∥∥t	VERB
ejpam-5587	279	9	2x	2x	NUM
ejpam-5587	279	10	∥∥	∥∥	X
ejpam-5587	279	11	∥x∥	∥x∥	NOUN
ejpam-5587	279	12	=	=	PUNCT
ejpam-5587	279	13	∥y∥	∥y∥	X
ejpam-5587	279	14	∥∥t−2y	∥∥t−2y	PROPN
ejpam-5587	279	15	∥∥	∥∥	X
ejpam-5587	279	16	.	.	PUNCT
ejpam-5587	280	1	hence	hence	ADV
ejpam-5587	280	2	t−1	t−1	PROPN
ejpam-5587	280	3	is	be	AUX
ejpam-5587	280	4	totally	totally	ADV
ejpam-5587	280	5	paranormal	paranormal	ADJ
ejpam-5587	280	6	since	since	SCONJ
ejpam-5587	280	7	totally	totally	ADV
ejpam-5587	280	8	paranormal	paranormal	PROPN
ejpam-5587	280	9	has	have	VERB
ejpam-5587	280	10	invariant	invariant	ADJ
ejpam-5587	280	11	translation	translation	NOUN
ejpam-5587	280	12	property	property	NOUN
ejpam-5587	280	13	.	.	PUNCT
ejpam-5587	281	1	(	(	PUNCT
ejpam-5587	281	2	iii	iii	X
ejpam-5587	281	3	)	)	PUNCT
ejpam-5587	281	4	suppose	suppose	VERB
ejpam-5587	281	5	on	on	ADP
ejpam-5587	281	6	the	the	DET
ejpam-5587	281	7	contrary	contrary	NOUN
ejpam-5587	281	8	that	that	SCONJ
ejpam-5587	281	9	σ(t	σ(t	PROPN
ejpam-5587	281	10	)	)	PUNCT
ejpam-5587	282	1	=	=	PUNCT
ejpam-5587	282	2	∅.	∅.	PROPN
ejpam-5587	282	3	then	then	ADV
ejpam-5587	282	4	t	t	PROPN
ejpam-5587	282	5	is	be	AUX
ejpam-5587	282	6	invertible	invertible	ADJ
ejpam-5587	282	7	and	and	CCONJ
ejpam-5587	282	8	t	t	NOUN
ejpam-5587	282	9	∈	∈	PROPN
ejpam-5587	282	10	b(h	b(h	PROPN
ejpam-5587	282	11	)	)	PUNCT
ejpam-5587	282	12	.	.	PUNCT
ejpam-5587	283	1	first	first	ADV
ejpam-5587	283	2	,	,	PUNCT
ejpam-5587	283	3	we	we	PRON
ejpam-5587	283	4	show	show	VERB
ejpam-5587	283	5	that	that	SCONJ
ejpam-5587	283	6	σ(t−1	σ(t−1	VERB
ejpam-5587	283	7	)	)	PUNCT
ejpam-5587	283	8	=	=	SYM
ejpam-5587	283	9	{	{	PUNCT
ejpam-5587	283	10	0	0	NUM
ejpam-5587	283	11	}	}	PUNCT
ejpam-5587	283	12	.	.	PUNCT
ejpam-5587	284	1	for	for	ADP
ejpam-5587	284	2	any	any	DET
ejpam-5587	284	3	complex	complex	ADJ
ejpam-5587	284	4	number	number	NOUN
ejpam-5587	284	5	µ	µ	ADP
ejpam-5587	284	6	̸=	̸=	PROPN
ejpam-5587	284	7	0	0	NUM
ejpam-5587	284	8	,	,	PUNCT
ejpam-5587	284	9	consider	consider	VERB
ejpam-5587	284	10	the	the	DET
ejpam-5587	284	11	operator	operator	NOUN
ejpam-5587	284	12	s	s	PART
ejpam-5587	284	13	=	=	PUNCT
ejpam-5587	284	14	µ−1(t	µ−1(t	ADV
ejpam-5587	284	15	−	−	PROPN
ejpam-5587	284	16	µ−1i)−1	µ−1i)−1	NOUN
ejpam-5587	284	17	.	.	PUNCT
ejpam-5587	285	1	here	here	ADV
ejpam-5587	285	2	s	s	VERB
ejpam-5587	285	3	can	can	AUX
ejpam-5587	285	4	also	also	ADV
ejpam-5587	285	5	be	be	AUX
ejpam-5587	285	6	written	write	VERB
ejpam-5587	285	7	as	as	ADP
ejpam-5587	285	8	the	the	DET
ejpam-5587	285	9	sum	sum	NOUN
ejpam-5587	285	10	of	of	ADP
ejpam-5587	285	11	two	two	NUM
ejpam-5587	285	12	bounded	bounded	ADJ
ejpam-5587	285	13	operators	operator	NOUN
ejpam-5587	285	14	,	,	PUNCT
ejpam-5587	285	15	s	s	PART
ejpam-5587	285	16	=	=	PUNCT
ejpam-5587	285	17	µ−1(i	µ−1(i	PROPN
ejpam-5587	285	18	+	+	NUM
ejpam-5587	285	19	µ−1(t	µ−1(t	ADV
ejpam-5587	285	20	−	−	PROPN
ejpam-5587	285	21	µ−1i)−1	µ−1i)−1	NOUN
ejpam-5587	285	22	)	)	PUNCT
ejpam-5587	285	23	,	,	PUNCT
ejpam-5587	285	24	so	so	CCONJ
ejpam-5587	285	25	s	s	VERB
ejpam-5587	285	26	is	be	AUX
ejpam-5587	285	27	bounded	bound	VERB
ejpam-5587	285	28	.	.	PUNCT
ejpam-5587	286	1	by	by	ADP
ejpam-5587	286	2	a	a	DET
ejpam-5587	286	3	simple	simple	ADJ
ejpam-5587	286	4	computation	computation	NOUN
ejpam-5587	286	5	we	we	PRON
ejpam-5587	286	6	can	can	AUX
ejpam-5587	286	7	show	show	VERB
ejpam-5587	286	8	that	that	SCONJ
ejpam-5587	286	9	s	s	VERB
ejpam-5587	286	10	is	be	AUX
ejpam-5587	286	11	the	the	DET
ejpam-5587	286	12	bounded	bounded	ADJ
ejpam-5587	286	13	inverse	inverse	NOUN
ejpam-5587	286	14	of	of	ADP
ejpam-5587	286	15	µi	µi	PROPN
ejpam-5587	286	16	−	−	PROPN
ejpam-5587	286	17	t−1	t−1	PROPN
ejpam-5587	286	18	.	.	PUNCT
ejpam-5587	287	1	thus	thus	ADV
ejpam-5587	287	2	σ(t−1	σ(t−1	VERB
ejpam-5587	287	3	)	)	PUNCT
ejpam-5587	287	4	⊆	⊆	NUM
ejpam-5587	287	5	{	{	PUNCT
ejpam-5587	287	6	0	0	NUM
ejpam-5587	287	7	}	}	PUNCT
ejpam-5587	287	8	.	.	PUNCT
ejpam-5587	288	1	as	as	ADP
ejpam-5587	288	2	t−1	t−1	PROPN
ejpam-5587	288	3	∈	∈	PROPN
ejpam-5587	288	4	b(h	b(h	PROPN
ejpam-5587	288	5	)	)	PUNCT
ejpam-5587	288	6	,	,	PUNCT
ejpam-5587	288	7	this	this	PRON
ejpam-5587	288	8	implies	imply	VERB
ejpam-5587	288	9	σ(t−1	σ(t−1	NOUN
ejpam-5587	288	10	)	)	PUNCT
ejpam-5587	288	11	is	be	AUX
ejpam-5587	288	12	non	non	ADJ
ejpam-5587	288	13	-	-	ADJ
ejpam-5587	288	14	empty	empty	ADJ
ejpam-5587	288	15	,	,	PUNCT
ejpam-5587	288	16	so	so	SCONJ
ejpam-5587	288	17	we	we	PRON
ejpam-5587	288	18	conclude	conclude	VERB
ejpam-5587	288	19	that	that	PRON
ejpam-5587	288	20	σ(t−1	σ(t−1	VERB
ejpam-5587	288	21	)	)	PUNCT
ejpam-5587	288	22	=	=	SYM
ejpam-5587	288	23	{	{	PUNCT
ejpam-5587	288	24	0	0	NUM
ejpam-5587	288	25	}	}	PUNCT
ejpam-5587	288	26	.	.	PUNCT
ejpam-5587	289	1	by	by	ADP
ejpam-5587	289	2	(	(	PUNCT
ejpam-5587	289	3	ii	ii	NOUN
ejpam-5587	289	4	)	)	PUNCT
ejpam-5587	289	5	,	,	PUNCT
ejpam-5587	289	6	t−1	t−1	PROPN
ejpam-5587	289	7	is	be	AUX
ejpam-5587	289	8	bounded	bound	VERB
ejpam-5587	289	9	totally	totally	ADV
ejpam-5587	289	10	paranormal	paranormal	ADJ
ejpam-5587	289	11	operator	operator	NOUN
ejpam-5587	289	12	and	and	CCONJ
ejpam-5587	289	13	consequently	consequently	ADV
ejpam-5587	289	14	normaloid	normaloid	ADP
ejpam-5587	289	15	by	by	ADP
ejpam-5587	289	16	theorem	theorem	NOUN
ejpam-5587	289	17	3	3	NUM
ejpam-5587	289	18	.	.	PUNCT
ejpam-5587	290	1	hence∥∥t−1	hence∥∥t−1	X
ejpam-5587	290	2	∥∥	∥∥	X
ejpam-5587	290	3	=	=	SYM
ejpam-5587	290	4	0	0	PROPN
ejpam-5587	290	5	,	,	PUNCT
ejpam-5587	290	6	which	which	PRON
ejpam-5587	290	7	implies	imply	VERB
ejpam-5587	290	8	t−1	t−1	PROPN
ejpam-5587	290	9	=	=	SYM
ejpam-5587	290	10	0	0	PROPN
ejpam-5587	290	11	,	,	PUNCT
ejpam-5587	290	12	a	a	DET
ejpam-5587	290	13	contradiction	contradiction	NOUN
ejpam-5587	290	14	.	.	PUNCT
ejpam-5587	291	1	hence	hence	ADV
ejpam-5587	291	2	σ(t	σ(t	PROPN
ejpam-5587	291	3	)	)	PUNCT
ejpam-5587	291	4	is	be	AUX
ejpam-5587	291	5	non	non	ADJ
ejpam-5587	291	6	-	-	ADJ
ejpam-5587	291	7	empty	empty	ADJ
ejpam-5587	291	8	.	.	PUNCT
ejpam-5587	292	1	now	now	ADV
ejpam-5587	292	2	we	we	PRON
ejpam-5587	292	3	discuss	discuss	VERB
ejpam-5587	292	4	about	about	ADP
ejpam-5587	292	5	isolated	isolated	ADJ
ejpam-5587	292	6	spectral	spectral	ADJ
ejpam-5587	292	7	values	value	NOUN
ejpam-5587	292	8	of	of	ADP
ejpam-5587	292	9	totally	totally	ADV
ejpam-5587	292	10	paranormal	paranormal	ADJ
ejpam-5587	292	11	operators	operator	NOUN
ejpam-5587	292	12	.	.	PUNCT
ejpam-5587	293	1	theorem	theorem	VERB
ejpam-5587	293	2	6	6	NUM
ejpam-5587	293	3	.	.	PUNCT
ejpam-5587	294	1	let	let	VERB
ejpam-5587	294	2	t	t	PROPN
ejpam-5587	294	3	be	be	AUX
ejpam-5587	294	4	a	a	DET
ejpam-5587	294	5	densely	densely	ADV
ejpam-5587	294	6	defined	define	VERB
ejpam-5587	294	7	closed	close	VERB
ejpam-5587	294	8	totally	totally	ADV
ejpam-5587	294	9	paranormal	paranormal	ADJ
ejpam-5587	294	10	operator	operator	NOUN
ejpam-5587	294	11	.	.	PUNCT
ejpam-5587	295	1	if	if	SCONJ
ejpam-5587	295	2	µ	µ	NOUN
ejpam-5587	295	3	is	be	AUX
ejpam-5587	295	4	an	an	DET
ejpam-5587	295	5	isolated	isolated	ADJ
ejpam-5587	295	6	point	point	NOUN
ejpam-5587	295	7	of	of	ADP
ejpam-5587	295	8	σ(t	σ(t	PROPN
ejpam-5587	295	9	)	)	PUNCT
ejpam-5587	295	10	,	,	PUNCT
ejpam-5587	295	11	then	then	ADV
ejpam-5587	295	12	ker(t	ker(t	NOUN
ejpam-5587	295	13	−	−	PROPN
ejpam-5587	295	14	µi	µi	PROPN
ejpam-5587	295	15	)	)	PUNCT
ejpam-5587	295	16	=	=	SYM
ejpam-5587	295	17	ran(eµ	ran(eµ	NOUN
ejpam-5587	295	18	)	)	PUNCT
ejpam-5587	295	19	.	.	PUNCT
ejpam-5587	296	1	proof	proof	NOUN
ejpam-5587	296	2	.	.	PUNCT
ejpam-5587	297	1	it	it	PRON
ejpam-5587	297	2	follows	follow	VERB
ejpam-5587	297	3	from	from	ADP
ejpam-5587	297	4	[	[	X
ejpam-5587	297	5	4	4	NUM
ejpam-5587	297	6	,	,	PUNCT
ejpam-5587	297	7	lemma	lemma	PROPN
ejpam-5587	297	8	3.4	3.4	NUM
ejpam-5587	297	9	]	]	PUNCT
ejpam-5587	297	10	that	that	PRON
ejpam-5587	297	11	ker(t	ker(t	NOUN
ejpam-5587	297	12	−	−	PROPN
ejpam-5587	297	13	µi	µi	PROPN
ejpam-5587	297	14	)	)	PUNCT
ejpam-5587	297	15	⊆	⊆	NUM
ejpam-5587	297	16	ran(eµ	ran(eµ	NOUN
ejpam-5587	297	17	)	)	PUNCT
ejpam-5587	297	18	.	.	PUNCT
ejpam-5587	298	1	to	to	PART
ejpam-5587	298	2	complete	complete	VERB
ejpam-5587	298	3	the	the	DET
ejpam-5587	298	4	proof	proof	NOUN
ejpam-5587	298	5	we	we	PRON
ejpam-5587	298	6	have	have	VERB
ejpam-5587	298	7	to	to	PART
ejpam-5587	298	8	show	show	VERB
ejpam-5587	298	9	that	that	DET
ejpam-5587	298	10	ker(t	ker(t	NOUN
ejpam-5587	298	11	−	−	PROPN
ejpam-5587	298	12	µi	µi	PROPN
ejpam-5587	298	13	)	)	PUNCT
ejpam-5587	298	14	⊇	⊇	PROPN
ejpam-5587	298	15	ran(eµ	ran(eµ	NOUN
ejpam-5587	298	16	)	)	PUNCT
ejpam-5587	298	17	.	.	PUNCT
ejpam-5587	299	1	as	as	ADP
ejpam-5587	299	2	a	a	DET
ejpam-5587	299	3	consequence	consequence	NOUN
ejpam-5587	299	4	of	of	ADP
ejpam-5587	299	5	theorem	theorem	ADJ
ejpam-5587	299	6	1	1	NUM
ejpam-5587	299	7	and	and	CCONJ
ejpam-5587	299	8	theorem	theorem	VERB
ejpam-5587	299	9	5	5	NUM
ejpam-5587	299	10	,	,	PUNCT
ejpam-5587	299	11	we	we	PRON
ejpam-5587	299	12	know	know	VERB
ejpam-5587	299	13	that	that	SCONJ
ejpam-5587	299	14	t	t	PROPN
ejpam-5587	299	15	|ran(eµ	|ran(eµ	PROPN
ejpam-5587	299	16	)	)	PUNCT
ejpam-5587	299	17	is	be	AUX
ejpam-5587	299	18	bounded	bound	VERB
ejpam-5587	299	19	s.	s.	PROPN
ejpam-5587	299	20	alnabulsi	alnabulsi	PROPN
ejpam-5587	299	21	,	,	PUNCT
ejpam-5587	299	22	m.h.m	m.h.m	PROPN
ejpam-5587	299	23	.	.	PUNCT
ejpam-5587	299	24	rashid	rashid	PROPN
ejpam-5587	299	25	/	/	SYM
ejpam-5587	299	26	eur	eur	PROPN
ejpam-5587	299	27	.	.	PUNCT
ejpam-5587	300	1	j.	j.	PROPN
ejpam-5587	300	2	pure	pure	PROPN
ejpam-5587	300	3	appl	appl	PROPN
ejpam-5587	300	4	.	.	PROPN
ejpam-5587	300	5	math	math	PROPN
ejpam-5587	300	6	,	,	PUNCT
ejpam-5587	300	7	18	18	NUM
ejpam-5587	300	8	(	(	PUNCT
ejpam-5587	300	9	1	1	NUM
ejpam-5587	300	10	)	)	PUNCT
ejpam-5587	300	11	(	(	PUNCT
ejpam-5587	300	12	2025	2025	NUM
ejpam-5587	300	13	)	)	PUNCT
ejpam-5587	300	14	,	,	PUNCT
ejpam-5587	300	15	5587	5587	NUM
ejpam-5587	300	16	10	10	NUM
ejpam-5587	300	17	of	of	ADP
ejpam-5587	300	18	20	20	NUM
ejpam-5587	300	19	and	and	CCONJ
ejpam-5587	300	20	totally	totally	ADV
ejpam-5587	300	21	paranormal	paranormal	ADJ
ejpam-5587	300	22	.	.	PUNCT
ejpam-5587	301	1	by	by	ADP
ejpam-5587	301	2	theorem	theorem	NOUN
ejpam-5587	301	3	3	3	NUM
ejpam-5587	301	4	it	it	PRON
ejpam-5587	301	5	follows	follow	VERB
ejpam-5587	301	6	that	that	SCONJ
ejpam-5587	301	7	t	t	PROPN
ejpam-5587	301	8	|ran(eµ	|ran(eµ	PROPN
ejpam-5587	301	9	)	)	PUNCT
ejpam-5587	301	10	is	be	AUX
ejpam-5587	301	11	normaloid	normaloid	NOUN
ejpam-5587	301	12	.	.	PUNCT
ejpam-5587	302	1	if	if	SCONJ
ejpam-5587	302	2	µ	µ	X
ejpam-5587	302	3	=	=	SYM
ejpam-5587	302	4	0	0	NUM
ejpam-5587	302	5	,	,	PUNCT
ejpam-5587	302	6	then	then	ADV
ejpam-5587	302	7	σ	σ	PROPN
ejpam-5587	302	8	(	(	PUNCT
ejpam-5587	302	9	t	t	PROPN
ejpam-5587	302	10	|ran(e0	|ran(e0	PROPN
ejpam-5587	302	11	)	)	PUNCT
ejpam-5587	302	12	)	)	PUNCT
ejpam-5587	303	1	=	=	PUNCT
ejpam-5587	303	2	{	{	PUNCT
ejpam-5587	303	3	0	0	NUM
ejpam-5587	303	4	}	}	PUNCT
ejpam-5587	303	5	.	.	PUNCT
ejpam-5587	304	1	this	this	PRON
ejpam-5587	304	2	implies	imply	VERB
ejpam-5587	304	3	∥∥t	∥∥t	VERB
ejpam-5587	304	4	|ran(e0	|ran(e0	NOUN
ejpam-5587	304	5	)	)	PUNCT
ejpam-5587	304	6	∥∥	∥∥	X
ejpam-5587	305	1	=	=	SYM
ejpam-5587	305	2	0	0	PUNCT
ejpam-5587	305	3	and	and	CCONJ
ejpam-5587	305	4	consequently	consequently	ADV
ejpam-5587	305	5	t	t	PRON
ejpam-5587	305	6	|ran(e0	|ran(e0	PROPN
ejpam-5587	305	7	)	)	PUNCT
ejpam-5587	305	8	=	=	SYM
ejpam-5587	305	9	0	0	X
ejpam-5587	305	10	.	.	PUNCT
ejpam-5587	305	11	hence	hence	ADV
ejpam-5587	305	12	ran(e0	ran(e0	NOUN
ejpam-5587	305	13	)	)	PUNCT
ejpam-5587	305	14	⊆	⊆	NUM
ejpam-5587	305	15	ker(t	ker(t	NOUN
ejpam-5587	305	16	)	)	PUNCT
ejpam-5587	305	17	.	.	PUNCT
ejpam-5587	306	1	if	if	SCONJ
ejpam-5587	306	2	µ	µ	DET
ejpam-5587	306	3	̸=	̸=	PROPN
ejpam-5587	306	4	0	0	NUM
ejpam-5587	306	5	,	,	PUNCT
ejpam-5587	306	6	then	then	ADV
ejpam-5587	306	7	σ	σ	PROPN
ejpam-5587	306	8	(	(	PUNCT
ejpam-5587	306	9	µ−1	µ−1	PROPN
ejpam-5587	306	10	t	t	PROPN
ejpam-5587	306	11	|ran(eµ	|ran(eµ	PROPN
ejpam-5587	306	12	)	)	PUNCT
ejpam-5587	306	13	)	)	PUNCT
ejpam-5587	307	1	=	=	PUNCT
ejpam-5587	308	1	{	{	PUNCT
ejpam-5587	308	2	1	1	NUM
ejpam-5587	308	3	}	}	PUNCT
ejpam-5587	308	4	.	.	PUNCT
ejpam-5587	309	1	by	by	ADP
ejpam-5587	309	2	theorem	theorem	NOUN
ejpam-5587	309	3	3	3	NUM
ejpam-5587	309	4	,	,	PUNCT
ejpam-5587	309	5	it	it	PRON
ejpam-5587	309	6	follows	follow	VERB
ejpam-5587	309	7	that	that	SCONJ
ejpam-5587	309	8	µ−1	µ−1	PROPN
ejpam-5587	309	9	t	t	PROPN
ejpam-5587	309	10	|ran(eµ	|ran(eµ	PROPN
ejpam-5587	309	11	)	)	PUNCT
ejpam-5587	309	12	is	be	AUX
ejpam-5587	309	13	unitary	unitary	ADJ
ejpam-5587	309	14	.	.	PUNCT
ejpam-5587	310	1	thus	thus	ADV
ejpam-5587	310	2	t	t	PROPN
ejpam-5587	310	3	|ran(eµ	|ran(eµ	PROPN
ejpam-5587	310	4	)	)	PUNCT
ejpam-5587	310	5	−	−	PROPN
ejpam-5587	310	6	µiran(eµ	µiran(eµ	NOUN
ejpam-5587	310	7	)	)	PUNCT
ejpam-5587	310	8	is	be	AUX
ejpam-5587	310	9	normal	normal	ADJ
ejpam-5587	310	10	and	and	CCONJ
ejpam-5587	310	11	σ	σ	PROPN
ejpam-5587	310	12	(	(	PUNCT
ejpam-5587	310	13	t	t	PROPN
ejpam-5587	310	14	|ran(eµ	|ran(eµ	PROPN
ejpam-5587	310	15	)	)	PUNCT
ejpam-5587	310	16	−	−	PROPN
ejpam-5587	310	17	µiran(eµ	µiran(eµ	NOUN
ejpam-5587	310	18	)	)	PUNCT
ejpam-5587	310	19	)	)	PUNCT
ejpam-5587	311	1	=	=	PUNCT
ejpam-5587	311	2	{	{	PUNCT
ejpam-5587	311	3	0	0	NUM
ejpam-5587	311	4	}	}	PUNCT
ejpam-5587	311	5	.	.	PUNCT
ejpam-5587	312	1	since	since	SCONJ
ejpam-5587	312	2	every	every	DET
ejpam-5587	312	3	normal	normal	ADJ
ejpam-5587	312	4	operator	operator	NOUN
ejpam-5587	312	5	is	be	AUX
ejpam-5587	312	6	normaloid	normaloid	NOUN
ejpam-5587	312	7	,	,	PUNCT
ejpam-5587	312	8	we	we	PRON
ejpam-5587	312	9	conclude	conclude	VERB
ejpam-5587	312	10	that	that	SCONJ
ejpam-5587	312	11	t	t	PROPN
ejpam-5587	312	12	|ran(eµ	|ran(eµ	PROPN
ejpam-5587	312	13	)	)	PUNCT
ejpam-5587	312	14	−	−	PROPN
ejpam-5587	312	15	µiran(eµ	µiran(eµ	NOUN
ejpam-5587	312	16	)	)	PUNCT
ejpam-5587	312	17	=	=	SYM
ejpam-5587	313	1	0	0	X
ejpam-5587	313	2	.	.	PUNCT
ejpam-5587	313	3	hence	hence	ADV
ejpam-5587	313	4	ran(eµ	ran(eµ	NOUN
ejpam-5587	313	5	)	)	PUNCT
ejpam-5587	313	6	⊆	⊆	NUM
ejpam-5587	313	7	ker	ker	NOUN
ejpam-5587	313	8	(	(	PUNCT
ejpam-5587	313	9	t	t	PROPN
ejpam-5587	313	10	−	−	PROPN
ejpam-5587	313	11	µi	µi	PROPN
ejpam-5587	313	12	)	)	PUNCT
ejpam-5587	313	13	.	.	PUNCT
ejpam-5587	314	1	theorem	theorem	ADJ
ejpam-5587	314	2	7	7	NUM
ejpam-5587	314	3	.	.	PUNCT
ejpam-5587	315	1	let	let	VERB
ejpam-5587	315	2	t	t	PROPN
ejpam-5587	315	3	∈	∈	PROPN
ejpam-5587	315	4	l(h	l(h	PROPN
ejpam-5587	315	5	)	)	PUNCT
ejpam-5587	315	6	be	be	AUX
ejpam-5587	315	7	a	a	DET
ejpam-5587	315	8	densely	densely	ADV
ejpam-5587	315	9	defined	define	VERB
ejpam-5587	315	10	closed	close	VERB
ejpam-5587	315	11	totally	totally	ADV
ejpam-5587	315	12	paranormal	paranormal	ADJ
ejpam-5587	315	13	operator	operator	NOUN
ejpam-5587	315	14	and	and	CCONJ
ejpam-5587	315	15	µ	µ	NOUN
ejpam-5587	315	16	be	be	AUX
ejpam-5587	315	17	an	an	DET
ejpam-5587	315	18	isolated	isolated	ADJ
ejpam-5587	315	19	point	point	NOUN
ejpam-5587	315	20	of	of	ADP
ejpam-5587	315	21	σ(t	σ(t	PROPN
ejpam-5587	315	22	)	)	PUNCT
ejpam-5587	315	23	.	.	PUNCT
ejpam-5587	316	1	then	then	ADV
ejpam-5587	316	2	ker(eµ	ker(eµ	X
ejpam-5587	316	3	)	)	PUNCT
ejpam-5587	317	1	=	=	SYM
ejpam-5587	317	2	ran(t	ran(t	ADJ
ejpam-5587	317	3	−	−	PROPN
ejpam-5587	317	4	µi	µi	PROPN
ejpam-5587	317	5	)	)	PUNCT
ejpam-5587	317	6	.	.	PUNCT
ejpam-5587	318	1	proof	proof	NOUN
ejpam-5587	318	2	.	.	PUNCT
ejpam-5587	319	1	by	by	ADP
ejpam-5587	319	2	theorem	theorem	NOUN
ejpam-5587	319	3	1	1	NUM
ejpam-5587	319	4	,	,	PUNCT
ejpam-5587	319	5	µ	µ	X
ejpam-5587	319	6	/∈	/∈	PUNCT
ejpam-5587	319	7	σ	σ	PROPN
ejpam-5587	319	8	(	(	PUNCT
ejpam-5587	319	9	t	t	PROPN
ejpam-5587	319	10	|ker(eµ	|ker(eµ	PROPN
ejpam-5587	319	11	)	)	PUNCT
ejpam-5587	319	12	)	)	PUNCT
ejpam-5587	319	13	.	.	PUNCT
ejpam-5587	320	1	this	this	PRON
ejpam-5587	320	2	implies	imply	VERB
ejpam-5587	320	3	that	that	SCONJ
ejpam-5587	320	4	ran(t	ran(t	PROPN
ejpam-5587	320	5	−µi)|ker(eµ	−µi)|ker(eµ	PROPN
ejpam-5587	320	6	)	)	PUNCT
ejpam-5587	320	7	=	=	SYM
ejpam-5587	320	8	n(eµ	n(eµ	PROPN
ejpam-5587	320	9	)	)	PUNCT
ejpam-5587	320	10	and	and	CCONJ
ejpam-5587	320	11	consequently	consequently	ADV
ejpam-5587	320	12	n(eµ	n(eµ	NUM
ejpam-5587	320	13	)	)	PUNCT
ejpam-5587	320	14	⊆	⊆	NUM
ejpam-5587	320	15	ran(t	ran(t	PROPN
ejpam-5587	320	16	−	−	PROPN
ejpam-5587	320	17	µi	µi	PROPN
ejpam-5587	320	18	)	)	PUNCT
ejpam-5587	320	19	.	.	PUNCT
ejpam-5587	321	1	let	let	VERB
ejpam-5587	321	2	y	y	PROPN
ejpam-5587	321	3	∈	∈	PROPN
ejpam-5587	321	4	ran(t	ran(t	PROPN
ejpam-5587	321	5	−	−	PROPN
ejpam-5587	321	6	µi	µi	PROPN
ejpam-5587	321	7	)	)	PUNCT
ejpam-5587	321	8	.	.	PUNCT
ejpam-5587	322	1	there	there	PRON
ejpam-5587	322	2	exist	exist	VERB
ejpam-5587	322	3	x	x	SYM
ejpam-5587	322	4	∈	∈	PROPN
ejpam-5587	322	5	d(t	d(t	PROPN
ejpam-5587	322	6	)	)	PUNCT
ejpam-5587	322	7	such	such	ADJ
ejpam-5587	322	8	that	that	SCONJ
ejpam-5587	322	9	y	y	PROPN
ejpam-5587	322	10	=	=	SYM
ejpam-5587	322	11	(	(	PUNCT
ejpam-5587	322	12	t	t	PROPN
ejpam-5587	322	13	−	−	PROPN
ejpam-5587	322	14	µi)x	µi)x	PROPN
ejpam-5587	322	15	.	.	PUNCT
ejpam-5587	323	1	since	since	SCONJ
ejpam-5587	323	2	h	h	NOUN
ejpam-5587	323	3	=	=	PUNCT
ejpam-5587	323	4	ran(eµ	ran(eµ	PROPN
ejpam-5587	323	5	)	)	PUNCT
ejpam-5587	323	6	+	+	NUM
ejpam-5587	323	7	ker(eµ	ker(eµ	NOUN
ejpam-5587	323	8	)	)	PUNCT
ejpam-5587	323	9	and	and	CCONJ
ejpam-5587	323	10	ran(eµ	ran(eµ	NOUN
ejpam-5587	323	11	)	)	PUNCT
ejpam-5587	323	12	∩	∩	ADJ
ejpam-5587	323	13	ker(eµ	ker(eµ	NOUN
ejpam-5587	323	14	)	)	PUNCT
ejpam-5587	323	15	=	=	PRON
ejpam-5587	323	16	{	{	PUNCT
ejpam-5587	323	17	0	0	NUM
ejpam-5587	323	18	}	}	PUNCT
ejpam-5587	323	19	,	,	PUNCT
ejpam-5587	323	20	we	we	PRON
ejpam-5587	323	21	have	have	VERB
ejpam-5587	323	22	x	x	X
ejpam-5587	324	1	=	=	PUNCT
ejpam-5587	324	2	p	p	PROPN
ejpam-5587	325	1	+	+	CCONJ
ejpam-5587	325	2	q	q	ADJ
ejpam-5587	325	3	,	,	PUNCT
ejpam-5587	325	4	where	where	SCONJ
ejpam-5587	325	5	p	p	PROPN
ejpam-5587	325	6	∈	∈	PROPN
ejpam-5587	325	7	ran(eµ	ran(eµ	NOUN
ejpam-5587	325	8	)	)	PUNCT
ejpam-5587	325	9	and	and	CCONJ
ejpam-5587	325	10	q	q	PROPN
ejpam-5587	325	11	∈	∈	PROPN
ejpam-5587	325	12	ker(eµ	ker(eµ	PROPN
ejpam-5587	325	13	)	)	PUNCT
ejpam-5587	325	14	.	.	PUNCT
ejpam-5587	326	1	it	it	PRON
ejpam-5587	326	2	follows	follow	VERB
ejpam-5587	326	3	from	from	ADP
ejpam-5587	326	4	theorem	theorem	ADJ
ejpam-5587	326	5	6	6	NUM
ejpam-5587	326	6	,	,	PUNCT
ejpam-5587	326	7	that	that	SCONJ
ejpam-5587	326	8	p	p	PROPN
ejpam-5587	326	9	∈	∈	PROPN
ejpam-5587	326	10	ker(t	ker(t	NOUN
ejpam-5587	326	11	−	−	PROPN
ejpam-5587	326	12	µi	µi	PROPN
ejpam-5587	326	13	)	)	PUNCT
ejpam-5587	326	14	⊆	⊆	PROPN
ejpam-5587	326	15	d(t	d(t	PROPN
ejpam-5587	326	16	)	)	PUNCT
ejpam-5587	326	17	and	and	CCONJ
ejpam-5587	326	18	consequently	consequently	ADV
ejpam-5587	326	19	q	q	X
ejpam-5587	327	1	=	=	PUNCT
ejpam-5587	327	2	x	x	X
ejpam-5587	327	3	−	−	PROPN
ejpam-5587	327	4	p	p	X
ejpam-5587	327	5	∈	∈	PROPN
ejpam-5587	327	6	d(t	d(t	PROPN
ejpam-5587	327	7	)	)	PUNCT
ejpam-5587	327	8	.	.	PUNCT
ejpam-5587	328	1	as	as	SCONJ
ejpam-5587	328	2	we	we	PRON
ejpam-5587	328	3	know	know	VERB
ejpam-5587	328	4	from	from	ADP
ejpam-5587	328	5	theorem	theorem	NOUN
ejpam-5587	328	6	1	1	NUM
ejpam-5587	328	7	that	that	SCONJ
ejpam-5587	328	8	ker(eµ	ker(eµ	NOUN
ejpam-5587	328	9	)	)	PUNCT
ejpam-5587	328	10	is	be	AUX
ejpam-5587	328	11	invariant	invariant	ADJ
ejpam-5587	328	12	under	under	ADP
ejpam-5587	328	13	t	t	PROPN
ejpam-5587	328	14	,	,	PUNCT
ejpam-5587	328	15	we	we	PRON
ejpam-5587	328	16	have	have	VERB
ejpam-5587	328	17	y	y	NOUN
ejpam-5587	328	18	=	=	SYM
ejpam-5587	328	19	(	(	PUNCT
ejpam-5587	328	20	t	t	NOUN
ejpam-5587	328	21	−	−	PROPN
ejpam-5587	328	22	µi)x	µi)x	PROPN
ejpam-5587	328	23	=	=	SYM
ejpam-5587	328	24	(	(	PUNCT
ejpam-5587	328	25	t	t	PROPN
ejpam-5587	328	26	−	−	PROPN
ejpam-5587	329	1	µi)q	µi)q	PROPN
ejpam-5587	330	1	∈	∈	PROPN
ejpam-5587	331	1	(	(	PUNCT
ejpam-5587	331	2	t	t	NOUN
ejpam-5587	331	3	−	−	PROPN
ejpam-5587	331	4	µi)(ker(eµ	µi)(ker(eµ	NUM
ejpam-5587	331	5	)	)	PUNCT
ejpam-5587	331	6	)	)	PUNCT
ejpam-5587	332	1	⊆	⊆	NUM
ejpam-5587	332	2	ker(eµ	ker(eµ	NOUN
ejpam-5587	332	3	)	)	PUNCT
ejpam-5587	332	4	.	.	PUNCT
ejpam-5587	333	1	hence	hence	ADV
ejpam-5587	333	2	ran(t	ran(t	PROPN
ejpam-5587	333	3	−	−	PROPN
ejpam-5587	333	4	µi	µi	PROPN
ejpam-5587	333	5	)	)	PUNCT
ejpam-5587	333	6	⊆	⊆	NUM
ejpam-5587	333	7	ker(eµ	ker(eµ	NOUN
ejpam-5587	333	8	)	)	PUNCT
ejpam-5587	333	9	.	.	PUNCT
ejpam-5587	334	1	this	this	PRON
ejpam-5587	334	2	proves	prove	VERB
ejpam-5587	334	3	the	the	DET
ejpam-5587	334	4	result	result	NOUN
ejpam-5587	334	5	.	.	PUNCT
ejpam-5587	335	1	the	the	DET
ejpam-5587	335	2	following	follow	VERB
ejpam-5587	335	3	results	result	NOUN
ejpam-5587	335	4	are	be	AUX
ejpam-5587	335	5	consequences	consequence	NOUN
ejpam-5587	335	6	of	of	ADP
ejpam-5587	335	7	theorem	theorem	NOUN
ejpam-5587	335	8	7	7	NUM
ejpam-5587	335	9	which	which	PRON
ejpam-5587	335	10	gives	give	VERB
ejpam-5587	335	11	a	a	DET
ejpam-5587	335	12	characterization	characterization	NOUN
ejpam-5587	335	13	for	for	ADP
ejpam-5587	335	14	closed	closed	ADJ
ejpam-5587	335	15	range	range	NOUN
ejpam-5587	335	16	totally	totally	ADV
ejpam-5587	335	17	paranormal	paranormal	ADJ
ejpam-5587	335	18	operators	operator	NOUN
ejpam-5587	335	19	.	.	PUNCT
ejpam-5587	336	1	corollary	corollary	ADJ
ejpam-5587	336	2	1	1	PROPN
ejpam-5587	336	3	.	.	PUNCT
ejpam-5587	336	4	suppose	suppose	VERB
ejpam-5587	336	5	t	t	PROPN
ejpam-5587	336	6	∈	∈	PROPN
ejpam-5587	336	7	l(h	l(h	PROPN
ejpam-5587	336	8	)	)	PUNCT
ejpam-5587	336	9	is	be	AUX
ejpam-5587	336	10	a	a	DET
ejpam-5587	336	11	densely	densely	ADV
ejpam-5587	336	12	defined	define	VERB
ejpam-5587	336	13	closed	close	VERB
ejpam-5587	336	14	totally	totally	ADV
ejpam-5587	336	15	paranormal	paranormal	ADJ
ejpam-5587	336	16	operator	operator	NOUN
ejpam-5587	336	17	.	.	PUNCT
ejpam-5587	337	1	if	if	SCONJ
ejpam-5587	337	2	0	0	NUM
ejpam-5587	337	3	is	be	AUX
ejpam-5587	337	4	an	an	DET
ejpam-5587	337	5	isolated	isolated	ADJ
ejpam-5587	337	6	point	point	NOUN
ejpam-5587	337	7	of	of	ADP
ejpam-5587	337	8	σ(t	σ(t	PROPN
ejpam-5587	337	9	)	)	PUNCT
ejpam-5587	337	10	,	,	PUNCT
ejpam-5587	337	11	then	then	ADV
ejpam-5587	337	12	ran(t	ran(t	PROPN
ejpam-5587	337	13	)	)	PUNCT
ejpam-5587	337	14	is	be	AUX
ejpam-5587	337	15	closed	close	VERB
ejpam-5587	337	16	.	.	PUNCT
ejpam-5587	338	1	in	in	ADP
ejpam-5587	338	2	general	general	ADJ
ejpam-5587	338	3	,	,	PUNCT
ejpam-5587	338	4	the	the	DET
ejpam-5587	338	5	converse	converse	NOUN
ejpam-5587	338	6	of	of	ADP
ejpam-5587	338	7	corollary	corollary	ADJ
ejpam-5587	338	8	1	1	NUM
ejpam-5587	338	9	is	be	AUX
ejpam-5587	338	10	not	not	PART
ejpam-5587	338	11	true	true	ADJ
ejpam-5587	338	12	.	.	PUNCT
ejpam-5587	339	1	we	we	PRON
ejpam-5587	339	2	have	have	VERB
ejpam-5587	339	3	the	the	DET
ejpam-5587	339	4	following	following	ADJ
ejpam-5587	339	5	example	example	NOUN
ejpam-5587	339	6	to	to	PART
ejpam-5587	339	7	illustrate	illustrate	VERB
ejpam-5587	339	8	this	this	PRON
ejpam-5587	339	9	.	.	PUNCT
ejpam-5587	340	1	example	example	NOUN
ejpam-5587	341	1	2	2	NUM
ejpam-5587	341	2	.	.	PUNCT
ejpam-5587	341	3	let	let	VERB
ejpam-5587	341	4	t	t	NOUN
ejpam-5587	341	5	:	:	PUNCT
ejpam-5587	341	6	ℓ2(n	ℓ2(n	ADJ
ejpam-5587	341	7	)	)	PUNCT
ejpam-5587	341	8	→	→	SYM
ejpam-5587	341	9	ℓ2(n	ℓ2(n	VERB
ejpam-5587	341	10	)	)	PUNCT
ejpam-5587	341	11	be	be	AUX
ejpam-5587	341	12	defined	define	VERB
ejpam-5587	341	13	by	by	ADP
ejpam-5587	341	14	t	t	PROPN
ejpam-5587	341	15	(	(	PUNCT
ejpam-5587	341	16	x1	x1	PROPN
ejpam-5587	341	17	,	,	PUNCT
ejpam-5587	341	18	x2	x2	PROPN
ejpam-5587	341	19	,	,	PUNCT
ejpam-5587	341	20	·	·	PUNCT
ejpam-5587	341	21	·	·	PUNCT
ejpam-5587	341	22	·	·	PUNCT
ejpam-5587	341	23	)	)	PUNCT
ejpam-5587	342	1	=	=	PUNCT
ejpam-5587	342	2	(	(	PUNCT
ejpam-5587	342	3	0	0	NUM
ejpam-5587	342	4	,	,	PUNCT
ejpam-5587	342	5	x1	x1	PROPN
ejpam-5587	342	6	,	,	PUNCT
ejpam-5587	342	7	x2	x2	PROPN
ejpam-5587	342	8	,	,	PUNCT
ejpam-5587	342	9	·	·	PUNCT
ejpam-5587	342	10	·	·	PUNCT
ejpam-5587	342	11	·	·	PUNCT
ejpam-5587	342	12	)	)	PUNCT
ejpam-5587	342	13	,	,	PUNCT
ejpam-5587	342	14	for	for	ADP
ejpam-5587	342	15	all	all	DET
ejpam-5587	342	16	(	(	PUNCT
ejpam-5587	342	17	xn	xn	X
ejpam-5587	342	18	)	)	PUNCT
ejpam-5587	342	19	∈	∈	PROPN
ejpam-5587	342	20	ℓ2(n	ℓ2(n	PROPN
ejpam-5587	342	21	)	)	PUNCT
ejpam-5587	342	22	then	then	ADV
ejpam-5587	342	23	σ(t	σ(t	PROPN
ejpam-5587	342	24	)	)	PUNCT
ejpam-5587	342	25	=	=	PUNCT
ejpam-5587	343	1	{	{	PUNCT
ejpam-5587	343	2	z	z	NOUN
ejpam-5587	343	3	∈	∈	PROPN
ejpam-5587	343	4	c	c	NOUN
ejpam-5587	343	5	:	:	PUNCT
ejpam-5587	343	6	|z|	|z|	VERB
ejpam-5587	343	7	≤	≤	NUM
ejpam-5587	343	8	1	1	NUM
ejpam-5587	343	9	}	}	PUNCT
ejpam-5587	343	10	,	,	PUNCT
ejpam-5587	343	11	ran(t	ran(t	X
ejpam-5587	343	12	)	)	PUNCT
ejpam-5587	343	13	=	=	PUNCT
ejpam-5587	344	1	ℓ2(n	ℓ2(n	ADJ
ejpam-5587	344	2	)	)	PUNCT
ejpam-5587	344	3	\	\	NOUN
ejpam-5587	344	4	span	span	NOUN
ejpam-5587	344	5	{	{	PUNCT
ejpam-5587	344	6	e1	e1	PROPN
ejpam-5587	344	7	}	}	PUNCT
ejpam-5587	344	8	.	.	PUNCT
ejpam-5587	345	1	here	here	ADV
ejpam-5587	345	2	ran(t	ran(t	X
ejpam-5587	345	3	)	)	PUNCT
ejpam-5587	345	4	is	be	AUX
ejpam-5587	345	5	closed	close	VERB
ejpam-5587	345	6	but	but	CCONJ
ejpam-5587	345	7	0	0	NUM
ejpam-5587	345	8	is	be	AUX
ejpam-5587	345	9	not	not	PART
ejpam-5587	345	10	an	an	DET
ejpam-5587	345	11	isolated	isolated	ADJ
ejpam-5587	345	12	point	point	NOUN
ejpam-5587	345	13	of	of	ADP
ejpam-5587	345	14	σ(t	σ(t	PROPN
ejpam-5587	345	15	)	)	PUNCT
ejpam-5587	345	16	.	.	PUNCT
ejpam-5587	346	1	clearly	clearly	ADV
ejpam-5587	346	2	,	,	PUNCT
ejpam-5587	346	3	t	t	PROPN
ejpam-5587	346	4	is	be	AUX
ejpam-5587	346	5	a	a	DET
ejpam-5587	346	6	totally	totally	ADV
ejpam-5587	346	7	paranormal	paranormal	ADJ
ejpam-5587	346	8	operator	operator	NOUN
ejpam-5587	346	9	.	.	PUNCT
ejpam-5587	347	1	next	next	ADJ
ejpam-5587	347	2	result	result	NOUN
ejpam-5587	347	3	gives	give	VERB
ejpam-5587	347	4	a	a	DET
ejpam-5587	347	5	sufficient	sufficient	ADJ
ejpam-5587	347	6	condition	condition	NOUN
ejpam-5587	347	7	under	under	ADP
ejpam-5587	347	8	which	which	PRON
ejpam-5587	347	9	the	the	DET
ejpam-5587	347	10	converse	converse	NOUN
ejpam-5587	347	11	of	of	ADP
ejpam-5587	347	12	corollary	corollary	ADJ
ejpam-5587	347	13	1	1	NUM
ejpam-5587	347	14	is	be	AUX
ejpam-5587	347	15	also	also	ADV
ejpam-5587	347	16	true	true	ADJ
ejpam-5587	347	17	.	.	PUNCT
ejpam-5587	348	1	theorem	theorem	ADJ
ejpam-5587	348	2	8	8	NUM
ejpam-5587	348	3	.	.	PUNCT
ejpam-5587	349	1	let	let	VERB
ejpam-5587	349	2	t	t	PROPN
ejpam-5587	349	3	∈	∈	PROPN
ejpam-5587	349	4	l(h	l(h	PROPN
ejpam-5587	349	5	)	)	PUNCT
ejpam-5587	349	6	be	be	AUX
ejpam-5587	349	7	a	a	DET
ejpam-5587	349	8	densely	densely	ADV
ejpam-5587	349	9	defined	define	VERB
ejpam-5587	349	10	closed	close	VERB
ejpam-5587	349	11	totally	totally	ADV
ejpam-5587	349	12	paranormal	paranormal	ADJ
ejpam-5587	349	13	operator	operator	NOUN
ejpam-5587	349	14	with	with	ADP
ejpam-5587	349	15	ker(t	ker(t	NOUN
ejpam-5587	349	16	)	)	PUNCT
ejpam-5587	350	1	=	=	SYM
ejpam-5587	350	2	ker(t	ker(t	NOUN
ejpam-5587	350	3	∗	∗	NOUN
ejpam-5587	350	4	)	)	PUNCT
ejpam-5587	350	5	and	and	CCONJ
ejpam-5587	350	6	0	0	NUM
ejpam-5587	350	7	∈	∈	PROPN
ejpam-5587	350	8	σ(t	σ(t	PROPN
ejpam-5587	350	9	)	)	PUNCT
ejpam-5587	350	10	.	.	PUNCT
ejpam-5587	351	1	then	then	ADV
ejpam-5587	351	2	0	0	NUM
ejpam-5587	351	3	is	be	AUX
ejpam-5587	351	4	an	an	DET
ejpam-5587	351	5	isolated	isolated	ADJ
ejpam-5587	351	6	point	point	NOUN
ejpam-5587	351	7	of	of	ADP
ejpam-5587	351	8	σ(t	σ(t	PROPN
ejpam-5587	351	9	)	)	PUNCT
ejpam-5587	352	1	if	if	SCONJ
ejpam-5587	352	2	and	and	CCONJ
ejpam-5587	352	3	only	only	ADV
ejpam-5587	352	4	if	if	SCONJ
ejpam-5587	352	5	ran(t	ran(t	NOUN
ejpam-5587	352	6	)	)	PUNCT
ejpam-5587	352	7	is	be	AUX
ejpam-5587	352	8	closed	close	VERB
ejpam-5587	352	9	.	.	PUNCT
ejpam-5587	353	1	s.	s.	PROPN
ejpam-5587	353	2	alnabulsi	alnabulsi	PROPN
ejpam-5587	353	3	,	,	PUNCT
ejpam-5587	353	4	m.h.m	m.h.m	PROPN
ejpam-5587	353	5	.	.	PUNCT
ejpam-5587	353	6	rashid	rashid	PROPN
ejpam-5587	353	7	/	/	SYM
ejpam-5587	353	8	eur	eur	PROPN
ejpam-5587	353	9	.	.	PUNCT
ejpam-5587	354	1	j.	j.	PROPN
ejpam-5587	354	2	pure	pure	PROPN
ejpam-5587	354	3	appl	appl	PROPN
ejpam-5587	354	4	.	.	PROPN
ejpam-5587	354	5	math	math	PROPN
ejpam-5587	354	6	,	,	PUNCT
ejpam-5587	354	7	18	18	NUM
ejpam-5587	354	8	(	(	PUNCT
ejpam-5587	354	9	1	1	NUM
ejpam-5587	354	10	)	)	PUNCT
ejpam-5587	354	11	(	(	PUNCT
ejpam-5587	354	12	2025	2025	NUM
ejpam-5587	354	13	)	)	PUNCT
ejpam-5587	354	14	,	,	PUNCT
ejpam-5587	354	15	5587	5587	NUM
ejpam-5587	354	16	11	11	NUM
ejpam-5587	354	17	of	of	ADP
ejpam-5587	354	18	20	20	NUM
ejpam-5587	354	19	proof	proof	NOUN
ejpam-5587	354	20	.	.	PUNCT
ejpam-5587	355	1	the	the	DET
ejpam-5587	355	2	necessary	necessary	ADJ
ejpam-5587	355	3	condition	condition	NOUN
ejpam-5587	355	4	follows	follow	VERB
ejpam-5587	355	5	from	from	ADP
ejpam-5587	355	6	corollary	corollary	ADJ
ejpam-5587	355	7	1	1	NUM
ejpam-5587	355	8	.	.	PUNCT
ejpam-5587	355	9	to	to	PART
ejpam-5587	355	10	prove	prove	VERB
ejpam-5587	355	11	the	the	DET
ejpam-5587	355	12	sufficient	sufficient	ADJ
ejpam-5587	355	13	condition	condition	NOUN
ejpam-5587	355	14	.	.	PUNCT
ejpam-5587	356	1	assume	assume	VERB
ejpam-5587	356	2	that	that	SCONJ
ejpam-5587	356	3	ran(t	ran(t	PROPN
ejpam-5587	356	4	)	)	PUNCT
ejpam-5587	356	5	is	be	AUX
ejpam-5587	356	6	closed	close	VERB
ejpam-5587	356	7	.	.	PUNCT
ejpam-5587	357	1	consider	consider	VERB
ejpam-5587	357	2	s0	s0	PROPN
ejpam-5587	357	3	=	=	SYM
ejpam-5587	357	4	t	t	PROPN
ejpam-5587	357	5	|ker(t	|ker(t	NOUN
ejpam-5587	357	6	)	)	PUNCT
ejpam-5587	358	1	⊥	⊥	NOUN
ejpam-5587	358	2	:	:	PUNCT
ejpam-5587	358	3	ker(t	ker(t	NOUN
ejpam-5587	358	4	)	)	PUNCT
ejpam-5587	358	5	⊥∩d(t	⊥∩d(t	NOUN
ejpam-5587	358	6	)	)	PUNCT
ejpam-5587	358	7	→	→	SYM
ejpam-5587	358	8	ker(t	ker(t	NOUN
ejpam-5587	358	9	)	)	PUNCT
ejpam-5587	358	10	⊥.	⊥.	PROPN
ejpam-5587	358	11	clearly	clearly	ADV
ejpam-5587	358	12	s0	s0	PROPN
ejpam-5587	358	13	is	be	AUX
ejpam-5587	358	14	injective	injective	ADJ
ejpam-5587	358	15	and	and	CCONJ
ejpam-5587	358	16	ran(s0	ran(s0	ADJ
ejpam-5587	358	17	)	)	PUNCT
ejpam-5587	358	18	=	=	SYM
ejpam-5587	358	19	ran(t	ran(t	PROPN
ejpam-5587	358	20	)	)	PUNCT
ejpam-5587	358	21	is	be	AUX
ejpam-5587	358	22	closed	close	VERB
ejpam-5587	358	23	.	.	PUNCT
ejpam-5587	359	1	also	also	ADV
ejpam-5587	359	2	ran(s0	ran(s0	ADJ
ejpam-5587	359	3	)	)	PUNCT
ejpam-5587	359	4	=	=	SYM
ejpam-5587	360	1	ker(t	ker(t	NOUN
ejpam-5587	360	2	∗)⊥	∗)⊥	NOUN
ejpam-5587	360	3	=	=	SYM
ejpam-5587	360	4	ker(t	ker(t	NOUN
ejpam-5587	360	5	)	)	PUNCT
ejpam-5587	360	6	⊥	⊥	NOUN
ejpam-5587	360	7	,	,	PUNCT
ejpam-5587	360	8	consequently	consequently	ADV
ejpam-5587	360	9	s0	s0	PROPN
ejpam-5587	360	10	is	be	AUX
ejpam-5587	360	11	bijective	bijective	ADJ
ejpam-5587	360	12	and	and	CCONJ
ejpam-5587	360	13	s−1	s−1	PROPN
ejpam-5587	360	14	0	0	NUM
ejpam-5587	360	15	∈	∈	NOUN
ejpam-5587	360	16	b(ker(t	b(ker(t	NOUN
ejpam-5587	360	17	)	)	PUNCT
ejpam-5587	360	18	⊥	⊥	NOUN
ejpam-5587	360	19	)	)	PUNCT
ejpam-5587	360	20	.	.	PUNCT
ejpam-5587	361	1	thus	thus	ADV
ejpam-5587	361	2	0	0	NUM
ejpam-5587	361	3	/∈	/∈	PUNCT
ejpam-5587	361	4	σ(s0	σ(s0	PROPN
ejpam-5587	361	5	)	)	PUNCT
ejpam-5587	361	6	.	.	PUNCT
ejpam-5587	362	1	applying	apply	VERB
ejpam-5587	362	2	[	[	X
ejpam-5587	362	3	2	2	NUM
ejpam-5587	362	4	,	,	PUNCT
ejpam-5587	362	5	theorem	theorem	VERB
ejpam-5587	362	6	5.4	5.4	NUM
ejpam-5587	362	7	,	,	PUNCT
ejpam-5587	362	8	page	page	NOUN
ejpam-5587	362	9	289	289	NUM
ejpam-5587	362	10	]	]	PUNCT
ejpam-5587	362	11	,	,	PUNCT
ejpam-5587	362	12	σ(t	σ(t	PROPN
ejpam-5587	362	13	)	)	PUNCT
ejpam-5587	363	1	⊆	⊆	X
ejpam-5587	363	2	{	{	PUNCT
ejpam-5587	363	3	0	0	NUM
ejpam-5587	363	4	}	}	PUNCT
ejpam-5587	363	5	∪	∪	ADP
ejpam-5587	363	6	σ(s0	σ(s0	NOUN
ejpam-5587	363	7	)	)	PUNCT
ejpam-5587	363	8	.	.	PUNCT
ejpam-5587	364	1	since	since	SCONJ
ejpam-5587	364	2	0	0	NUM
ejpam-5587	364	3	∈	∈	PROPN
ejpam-5587	364	4	σ(t	σ(t	PROPN
ejpam-5587	364	5	)	)	PUNCT
ejpam-5587	364	6	,	,	PUNCT
ejpam-5587	364	7	we	we	PRON
ejpam-5587	364	8	have	have	VERB
ejpam-5587	364	9	σ(t	σ(t	NOUN
ejpam-5587	364	10	)	)	PUNCT
ejpam-5587	365	1	=	=	PUNCT
ejpam-5587	365	2	{	{	PUNCT
ejpam-5587	365	3	0	0	NUM
ejpam-5587	365	4	}	}	PUNCT
ejpam-5587	365	5	∪	∪	X
ejpam-5587	365	6	σ(s0	σ(s0	PROPN
ejpam-5587	365	7	)	)	PUNCT
ejpam-5587	365	8	and	and	CCONJ
ejpam-5587	365	9	hence	hence	ADV
ejpam-5587	365	10	0	0	NUM
ejpam-5587	365	11	is	be	AUX
ejpam-5587	365	12	an	an	DET
ejpam-5587	365	13	isolated	isolated	ADJ
ejpam-5587	365	14	point	point	NOUN
ejpam-5587	365	15	of	of	ADP
ejpam-5587	365	16	σ(t	σ(t	PROPN
ejpam-5587	365	17	)	)	PUNCT
ejpam-5587	365	18	.	.	PUNCT
ejpam-5587	366	1	note	note	VERB
ejpam-5587	366	2	that	that	SCONJ
ejpam-5587	366	3	theorem	theorem	VERB
ejpam-5587	366	4	8	8	NUM
ejpam-5587	366	5	does	do	AUX
ejpam-5587	366	6	not	not	PART
ejpam-5587	366	7	hold	hold	VERB
ejpam-5587	366	8	if	if	SCONJ
ejpam-5587	366	9	we	we	PRON
ejpam-5587	366	10	drop	drop	VERB
ejpam-5587	366	11	the	the	DET
ejpam-5587	366	12	condition	condition	NOUN
ejpam-5587	366	13	ker(t	ker(t	NOUN
ejpam-5587	366	14	)	)	PUNCT
ejpam-5587	367	1	=	=	SYM
ejpam-5587	367	2	ker(t	ker(t	NOUN
ejpam-5587	367	3	∗	∗	NOUN
ejpam-5587	367	4	)	)	PUNCT
ejpam-5587	367	5	.	.	PUNCT
ejpam-5587	368	1	consider	consider	VERB
ejpam-5587	368	2	the	the	DET
ejpam-5587	368	3	operator	operator	NOUN
ejpam-5587	368	4	t	t	PROPN
ejpam-5587	368	5	defined	define	VERB
ejpam-5587	368	6	in	in	ADP
ejpam-5587	368	7	example	example	NOUN
ejpam-5587	368	8	2	2	X
ejpam-5587	368	9	.	.	PUNCT
ejpam-5587	368	10	clearly	clearly	ADV
ejpam-5587	368	11	ker(t	ker(t	NOUN
ejpam-5587	368	12	)	)	PUNCT
ejpam-5587	369	1	=	=	PUNCT
ejpam-5587	369	2	{	{	PUNCT
ejpam-5587	369	3	0	0	NUM
ejpam-5587	369	4	}	}	PUNCT
ejpam-5587	369	5	̸=	̸=	PROPN
ejpam-5587	369	6	span{e1	span{e1	NOUN
ejpam-5587	369	7	}	}	PUNCT
ejpam-5587	369	8	=	=	SYM
ejpam-5587	369	9	ker(t	ker(t	NOUN
ejpam-5587	369	10	∗	∗	NOUN
ejpam-5587	369	11	)	)	PUNCT
ejpam-5587	369	12	,	,	PUNCT
ejpam-5587	369	13	and	and	CCONJ
ejpam-5587	369	14	ran(t	ran(t	NOUN
ejpam-5587	369	15	)	)	PUNCT
ejpam-5587	369	16	is	be	AUX
ejpam-5587	369	17	closed	close	VERB
ejpam-5587	369	18	but	but	CCONJ
ejpam-5587	369	19	0	0	NUM
ejpam-5587	369	20	is	be	AUX
ejpam-5587	369	21	not	not	PART
ejpam-5587	369	22	an	an	DET
ejpam-5587	369	23	isolated	isolated	ADJ
ejpam-5587	369	24	point	point	NOUN
ejpam-5587	369	25	of	of	ADP
ejpam-5587	369	26	σ(t	σ(t	PROPN
ejpam-5587	369	27	)	)	PUNCT
ejpam-5587	369	28	.	.	PUNCT
ejpam-5587	370	1	theorem	theorem	NOUN
ejpam-5587	370	2	9	9	NUM
ejpam-5587	370	3	.	.	PUNCT
ejpam-5587	371	1	let	let	VERB
ejpam-5587	371	2	t	t	PROPN
ejpam-5587	371	3	∈	∈	PROPN
ejpam-5587	371	4	l(h	l(h	PROPN
ejpam-5587	371	5	)	)	PUNCT
ejpam-5587	371	6	be	be	AUX
ejpam-5587	371	7	a	a	DET
ejpam-5587	371	8	densely	densely	ADV
ejpam-5587	371	9	defined	define	VERB
ejpam-5587	371	10	closed	close	VERB
ejpam-5587	371	11	totally	totally	ADV
ejpam-5587	371	12	paranormal	paranormal	ADJ
ejpam-5587	371	13	operator	operator	NOUN
ejpam-5587	371	14	.	.	PUNCT
ejpam-5587	372	1	if	if	SCONJ
ejpam-5587	372	2	ker(t	ker(t	NOUN
ejpam-5587	372	3	)	)	PUNCT
ejpam-5587	372	4	=	=	SYM
ejpam-5587	373	1	ker(t	ker(t	NOUN
ejpam-5587	373	2	∗	∗	NOUN
ejpam-5587	373	3	)	)	PUNCT
ejpam-5587	373	4	,	,	PUNCT
ejpam-5587	373	5	then	then	ADV
ejpam-5587	373	6	m(t	m(t	NOUN
ejpam-5587	373	7	)	)	PUNCT
ejpam-5587	373	8	=	=	PUNCT
ejpam-5587	373	9	d(0	d(0	PROPN
ejpam-5587	373	10	,	,	PUNCT
ejpam-5587	373	11	σ(t	σ(t	PROPN
ejpam-5587	373	12	)	)	PUNCT
ejpam-5587	373	13	)	)	PUNCT
ejpam-5587	373	14	,	,	PUNCT
ejpam-5587	373	15	the	the	DET
ejpam-5587	373	16	distance	distance	NOUN
ejpam-5587	373	17	between	between	ADP
ejpam-5587	373	18	0	0	NUM
ejpam-5587	373	19	and	and	CCONJ
ejpam-5587	373	20	σ(t	σ(t	PROPN
ejpam-5587	373	21	)	)	PUNCT
ejpam-5587	373	22	.	.	PUNCT
ejpam-5587	374	1	proof	proof	NOUN
ejpam-5587	374	2	.	.	PUNCT
ejpam-5587	375	1	we	we	PRON
ejpam-5587	375	2	will	will	AUX
ejpam-5587	375	3	prove	prove	VERB
ejpam-5587	375	4	this	this	DET
ejpam-5587	375	5	result	result	NOUN
ejpam-5587	375	6	by	by	ADP
ejpam-5587	375	7	considering	consider	VERB
ejpam-5587	375	8	the	the	DET
ejpam-5587	375	9	following	follow	VERB
ejpam-5587	375	10	two	two	NUM
ejpam-5587	375	11	cases	case	NOUN
ejpam-5587	375	12	,	,	PUNCT
ejpam-5587	375	13	which	which	PRON
ejpam-5587	375	14	exhaust	exhaust	VERB
ejpam-5587	375	15	all	all	DET
ejpam-5587	375	16	the	the	DET
ejpam-5587	375	17	possibilities	possibility	NOUN
ejpam-5587	375	18	.	.	PUNCT
ejpam-5587	376	1	case	case	NOUN
ejpam-5587	376	2	(	(	PUNCT
ejpam-5587	376	3	1	1	NUM
ejpam-5587	376	4	):	):	PUNCT
ejpam-5587	376	5	t	t	PROPN
ejpam-5587	376	6	is	be	AUX
ejpam-5587	376	7	not	not	PART
ejpam-5587	376	8	injective	injective	ADJ
ejpam-5587	376	9	.	.	PUNCT
ejpam-5587	377	1	clearly	clearly	ADV
ejpam-5587	377	2	m(t	m(t	NOUN
ejpam-5587	377	3	)	)	PUNCT
ejpam-5587	378	1	=	=	SYM
ejpam-5587	378	2	0	0	NUM
ejpam-5587	378	3	and	and	CCONJ
ejpam-5587	378	4	0	0	NUM
ejpam-5587	378	5	∈	∈	NOUN
ejpam-5587	378	6	σp(t	σp(t	PUNCT
ejpam-5587	378	7	)	)	PUNCT
ejpam-5587	378	8	.	.	PUNCT
ejpam-5587	379	1	hence	hence	ADV
ejpam-5587	379	2	m(t	m(t	NOUN
ejpam-5587	379	3	)	)	PUNCT
ejpam-5587	380	1	=	=	SYM
ejpam-5587	380	2	0	0	X
ejpam-5587	380	3	=	=	SYM
ejpam-5587	380	4	d(0	d(0	PROPN
ejpam-5587	380	5	,	,	PUNCT
ejpam-5587	380	6	σ(t	σ(t	PROPN
ejpam-5587	380	7	)	)	PUNCT
ejpam-5587	380	8	)	)	PUNCT
ejpam-5587	380	9	.	.	PUNCT
ejpam-5587	381	1	case	case	NOUN
ejpam-5587	381	2	(	(	PUNCT
ejpam-5587	381	3	2	2	NUM
ejpam-5587	381	4	):	):	PUNCT
ejpam-5587	381	5	t	t	PROPN
ejpam-5587	381	6	is	be	AUX
ejpam-5587	381	7	injective	injective	ADJ
ejpam-5587	381	8	.	.	PUNCT
ejpam-5587	382	1	it	it	PRON
ejpam-5587	382	2	suffices	suffice	VERB
ejpam-5587	382	3	to	to	PART
ejpam-5587	382	4	show	show	VERB
ejpam-5587	382	5	that	that	SCONJ
ejpam-5587	382	6	γ(t	γ(t	NOUN
ejpam-5587	382	7	)	)	PUNCT
ejpam-5587	383	1	=	=	SYM
ejpam-5587	383	2	d(0	d(0	NOUN
ejpam-5587	383	3	,	,	PUNCT
ejpam-5587	383	4	σ(t	σ(t	PROPN
ejpam-5587	383	5	)	)	PUNCT
ejpam-5587	383	6	)	)	PUNCT
ejpam-5587	383	7	because	because	SCONJ
ejpam-5587	383	8	m(t	m(t	NOUN
ejpam-5587	383	9	)	)	PUNCT
ejpam-5587	384	1	=	=	PUNCT
ejpam-5587	384	2	γ(t	γ(t	NOUN
ejpam-5587	384	3	)	)	PUNCT
ejpam-5587	384	4	.	.	PUNCT
ejpam-5587	385	1	first	first	ADV
ejpam-5587	385	2	assume	assume	VERB
ejpam-5587	385	3	that	that	SCONJ
ejpam-5587	385	4	γ(t	γ(t	NOUN
ejpam-5587	385	5	)	)	PUNCT
ejpam-5587	386	1	=	=	PUNCT
ejpam-5587	386	2	0	0	X
ejpam-5587	386	3	.	.	PUNCT
ejpam-5587	387	1	it	it	PRON
ejpam-5587	387	2	follows	follow	VERB
ejpam-5587	387	3	from	from	ADP
ejpam-5587	387	4	theorem	theorem	ADJ
ejpam-5587	387	5	2	2	NUM
ejpam-5587	387	6	that	that	PRON
ejpam-5587	387	7	ran(t	ran(t	NOUN
ejpam-5587	387	8	)	)	PUNCT
ejpam-5587	387	9	is	be	AUX
ejpam-5587	387	10	not	not	PART
ejpam-5587	387	11	closed	closed	ADJ
ejpam-5587	387	12	and	and	CCONJ
ejpam-5587	387	13	consequently	consequently	ADV
ejpam-5587	387	14	0	0	NUM
ejpam-5587	387	15	∈	∈	NOUN
ejpam-5587	387	16	σc(t	σc(t	NUM
ejpam-5587	387	17	)	)	PUNCT
ejpam-5587	387	18	.	.	PUNCT
ejpam-5587	388	1	thus	thus	ADV
ejpam-5587	388	2	d(0	d(0	PROPN
ejpam-5587	388	3	,	,	PUNCT
ejpam-5587	388	4	σ(t	σ(t	PROPN
ejpam-5587	388	5	)	)	PUNCT
ejpam-5587	388	6	)	)	PUNCT
ejpam-5587	389	1	=	=	SYM
ejpam-5587	389	2	0	0	PUNCT
ejpam-5587	390	1	=	=	SYM
ejpam-5587	390	2	γ(t	γ(t	NOUN
ejpam-5587	390	3	)	)	PUNCT
ejpam-5587	390	4	.	.	PUNCT
ejpam-5587	391	1	now	now	ADV
ejpam-5587	391	2	assume	assume	VERB
ejpam-5587	391	3	that	that	SCONJ
ejpam-5587	391	4	γ(t	γ(t	NOUN
ejpam-5587	391	5	)	)	PUNCT
ejpam-5587	391	6	>	>	X
ejpam-5587	392	1	0	0	X
ejpam-5587	392	2	.	.	PUNCT
ejpam-5587	393	1	as	as	ADP
ejpam-5587	393	2	a	a	DET
ejpam-5587	393	3	consequence	consequence	NOUN
ejpam-5587	393	4	of	of	ADP
ejpam-5587	393	5	theorem	theorem	NOUN
ejpam-5587	393	6	2	2	NUM
ejpam-5587	393	7	,	,	PUNCT
ejpam-5587	393	8	ran(t	ran(t	PROPN
ejpam-5587	393	9	)	)	PUNCT
ejpam-5587	393	10	is	be	AUX
ejpam-5587	393	11	closed	close	VERB
ejpam-5587	393	12	.	.	PUNCT
ejpam-5587	394	1	note	note	VERB
ejpam-5587	394	2	that	that	SCONJ
ejpam-5587	394	3	0	0	NUM
ejpam-5587	394	4	/∈	/∈	PUNCT
ejpam-5587	394	5	σ(t	σ(t	PROPN
ejpam-5587	394	6	)	)	PUNCT
ejpam-5587	394	7	,	,	PUNCT
ejpam-5587	394	8	otherwise	otherwise	ADV
ejpam-5587	394	9	theorem	theorem	VERB
ejpam-5587	394	10	9	9	NUM
ejpam-5587	394	11	and	and	CCONJ
ejpam-5587	394	12	theorem	theorem	VERB
ejpam-5587	394	13	6	6	NUM
ejpam-5587	394	14	implies	imply	VERB
ejpam-5587	394	15	that	that	SCONJ
ejpam-5587	394	16	0	0	NUM
ejpam-5587	394	17	∈	∈	NOUN
ejpam-5587	394	18	σp(t	σp(t	PUNCT
ejpam-5587	394	19	)	)	PUNCT
ejpam-5587	394	20	.	.	PUNCT
ejpam-5587	395	1	but	but	CCONJ
ejpam-5587	395	2	this	this	PRON
ejpam-5587	395	3	is	be	AUX
ejpam-5587	395	4	not	not	PART
ejpam-5587	395	5	true	true	ADJ
ejpam-5587	395	6	,	,	PUNCT
ejpam-5587	395	7	as	as	SCONJ
ejpam-5587	395	8	t	t	PROPN
ejpam-5587	395	9	is	be	AUX
ejpam-5587	395	10	injective	injective	ADJ
ejpam-5587	395	11	.	.	PUNCT
ejpam-5587	396	1	thus	thus	ADV
ejpam-5587	396	2	0	0	NUM
ejpam-5587	396	3	/∈	/∈	PUNCT
ejpam-5587	396	4	σ(t	σ(t	PROPN
ejpam-5587	396	5	)	)	PUNCT
ejpam-5587	396	6	and	and	CCONJ
ejpam-5587	396	7	t−1	t−1	PROPN
ejpam-5587	396	8	is	be	AUX
ejpam-5587	396	9	bounded	bound	VERB
ejpam-5587	396	10	totally	totally	ADV
ejpam-5587	396	11	paranormal	paranormal	ADJ
ejpam-5587	396	12	operator	operator	NOUN
ejpam-5587	396	13	,	,	PUNCT
ejpam-5587	396	14	by	by	ADP
ejpam-5587	396	15	theorem	theorem	NOUN
ejpam-5587	396	16	5	5	NUM
ejpam-5587	396	17	.	.	PUNCT
ejpam-5587	396	18	hence	hence	ADV
ejpam-5587	396	19	t−1	t−1	PROPN
ejpam-5587	396	20	is	be	AUX
ejpam-5587	396	21	normaloid	normaloid	NOUN
ejpam-5587	396	22	and	and	CCONJ
ejpam-5587	396	23	[	[	X
ejpam-5587	396	24	17	17	NUM
ejpam-5587	396	25	,	,	PUNCT
ejpam-5587	396	26	proposition	proposition	NOUN
ejpam-5587	396	27	2.12	2.12	NUM
ejpam-5587	396	28	]	]	PUNCT
ejpam-5587	396	29	implies	imply	VERB
ejpam-5587	396	30	that	that	SCONJ
ejpam-5587	396	31	γ(t	γ(t	NOUN
ejpam-5587	396	32	)	)	PUNCT
ejpam-5587	397	1	=	=	SYM
ejpam-5587	397	2	1	1	NUM
ejpam-5587	397	3	∥t−1∥	∥t−1∥	NUM
ejpam-5587	397	4	=	=	SYM
ejpam-5587	397	5	1	1	NUM
ejpam-5587	397	6	r(t−1	r(t−1	ADJ
ejpam-5587	397	7	)	)	PUNCT
ejpam-5587	397	8	=	=	SYM
ejpam-5587	397	9	1	1	NUM
ejpam-5587	397	10	sup	sup	NOUN
ejpam-5587	397	11	{	{	PUNCT
ejpam-5587	397	12	|µ|	|µ|	PROPN
ejpam-5587	397	13	:	:	PUNCT
ejpam-5587	397	14	µ	µ	PROPN
ejpam-5587	397	15	∈	∈	PROPN
ejpam-5587	397	16	σ(t−1	σ(t−1	NOUN
ejpam-5587	397	17	)	)	PUNCT
ejpam-5587	397	18	}	}	PUNCT
ejpam-5587	397	19	=	=	SYM
ejpam-5587	397	20	inf	inf	NOUN
ejpam-5587	397	21	{	{	PUNCT
ejpam-5587	397	22	|ν|	|ν|	ADV
ejpam-5587	397	23	:	:	PUNCT
ejpam-5587	397	24	ν	ν	X
ejpam-5587	397	25	∈	∈	PROPN
ejpam-5587	397	26	σ(t	σ(t	PROPN
ejpam-5587	397	27	)	)	PUNCT
ejpam-5587	397	28	}	}	PUNCT
ejpam-5587	397	29	=	=	SYM
ejpam-5587	397	30	d(0	d(0	NOUN
ejpam-5587	397	31	,	,	PUNCT
ejpam-5587	397	32	σ(t	σ(t	PROPN
ejpam-5587	397	33	)	)	PUNCT
ejpam-5587	397	34	)	)	PUNCT
ejpam-5587	397	35	.	.	PUNCT
ejpam-5587	398	1	this	this	PRON
ejpam-5587	398	2	completes	complete	VERB
ejpam-5587	398	3	the	the	DET
ejpam-5587	398	4	proof	proof	NOUN
ejpam-5587	398	5	.	.	PUNCT
ejpam-5587	399	1	as	as	ADP
ejpam-5587	399	2	a	a	DET
ejpam-5587	399	3	consequence	consequence	NOUN
ejpam-5587	399	4	of	of	ADP
ejpam-5587	399	5	theorem	theorem	NOUN
ejpam-5587	399	6	9	9	NUM
ejpam-5587	399	7	we	we	PRON
ejpam-5587	399	8	have	have	VERB
ejpam-5587	399	9	the	the	DET
ejpam-5587	399	10	following	follow	VERB
ejpam-5587	399	11	result	result	NOUN
ejpam-5587	399	12	.	.	PUNCT
ejpam-5587	400	1	corollary	corollary	ADJ
ejpam-5587	400	2	2	2	NUM
ejpam-5587	400	3	.	.	PUNCT
ejpam-5587	401	1	if	if	SCONJ
ejpam-5587	401	2	t	t	PROPN
ejpam-5587	401	3	∈	∈	PROPN
ejpam-5587	401	4	l(h	l(h	PROPN
ejpam-5587	401	5	)	)	PUNCT
ejpam-5587	401	6	is	be	AUX
ejpam-5587	401	7	a	a	DET
ejpam-5587	401	8	densely	densely	ADV
ejpam-5587	401	9	defined	define	VERB
ejpam-5587	401	10	closed	close	VERB
ejpam-5587	401	11	totally	totally	ADV
ejpam-5587	401	12	paranormal	paranormal	ADJ
ejpam-5587	401	13	operator	operator	NOUN
ejpam-5587	401	14	and	and	CCONJ
ejpam-5587	401	15	ker(t	ker(t	NOUN
ejpam-5587	401	16	)	)	PUNCT
ejpam-5587	402	1	=	=	SYM
ejpam-5587	402	2	ker(t	ker(t	NOUN
ejpam-5587	402	3	∗	∗	NOUN
ejpam-5587	402	4	)	)	PUNCT
ejpam-5587	402	5	,	,	PUNCT
ejpam-5587	402	6	then	then	ADV
ejpam-5587	402	7	γ(t	γ(t	NOUN
ejpam-5587	402	8	)	)	PUNCT
ejpam-5587	403	1	=	=	SYM
ejpam-5587	403	2	d(t	d(t	PROPN
ejpam-5587	403	3	)	)	PUNCT
ejpam-5587	403	4	:	:	PUNCT
ejpam-5587	403	5	=	=	SYM
ejpam-5587	403	6	inf	inf	PROPN
ejpam-5587	403	7	{	{	PUNCT
ejpam-5587	403	8	|µ|	|µ|	PROPN
ejpam-5587	403	9	:	:	PUNCT
ejpam-5587	403	10	µ	µ	X
ejpam-5587	403	11	∈	∈	NOUN
ejpam-5587	403	12	σ(t	σ(t	PROPN
ejpam-5587	403	13	)	)	PUNCT
ejpam-5587	403	14	\	\	PROPN
ejpam-5587	403	15	{	{	PUNCT
ejpam-5587	403	16	0	0	NUM
ejpam-5587	403	17	}	}	PUNCT
ejpam-5587	403	18	}	}	PUNCT
ejpam-5587	403	19	.	.	PUNCT
ejpam-5587	404	1	proof	proof	NOUN
ejpam-5587	404	2	.	.	PUNCT
ejpam-5587	405	1	consider	consider	VERB
ejpam-5587	405	2	the	the	DET
ejpam-5587	405	3	operator	operator	NOUN
ejpam-5587	405	4	s0	s0	NOUN
ejpam-5587	405	5	=	=	SYM
ejpam-5587	405	6	t	t	PROPN
ejpam-5587	405	7	|ker(t	|ker(t	NOUN
ejpam-5587	405	8	)	)	PUNCT
ejpam-5587	406	1	⊥	⊥	NOUN
ejpam-5587	406	2	:	:	PUNCT
ejpam-5587	406	3	d(t	d(t	PROPN
ejpam-5587	406	4	)	)	PUNCT
ejpam-5587	406	5	∩ker(t	∩ker(t	NUM
ejpam-5587	406	6	)	)	PUNCT
ejpam-5587	406	7	⊥	⊥	X
ejpam-5587	406	8	→	→	SYM
ejpam-5587	406	9	ker(t	ker(t	NOUN
ejpam-5587	406	10	)	)	PUNCT
ejpam-5587	406	11	⊥.	⊥.	NUM
ejpam-5587	406	12	by	by	ADP
ejpam-5587	406	13	theorem	theorem	NOUN
ejpam-5587	406	14	5	5	NUM
ejpam-5587	406	15	and	and	CCONJ
ejpam-5587	406	16	theorem	theorem	VERB
ejpam-5587	406	17	9	9	NUM
ejpam-5587	406	18	,	,	PUNCT
ejpam-5587	406	19	s0	s0	PROPN
ejpam-5587	406	20	is	be	AUX
ejpam-5587	406	21	paranormal	paranormal	ADJ
ejpam-5587	406	22	and	and	CCONJ
ejpam-5587	406	23	γ(t	γ(t	NOUN
ejpam-5587	406	24	)	)	PUNCT
ejpam-5587	407	1	=	=	SYM
ejpam-5587	407	2	m(s0	m(s0	NOUN
ejpam-5587	407	3	)	)	PUNCT
ejpam-5587	407	4	=	=	PUNCT
ejpam-5587	407	5	d(0	d(0	NOUN
ejpam-5587	407	6	,	,	PUNCT
ejpam-5587	407	7	σ(s0	σ(s0	PROPN
ejpam-5587	407	8	)	)	PUNCT
ejpam-5587	407	9	)	)	PUNCT
ejpam-5587	408	1	=	=	SYM
ejpam-5587	408	2	d(t	d(t	PROPN
ejpam-5587	408	3	)	)	PUNCT
ejpam-5587	408	4	.	.	PUNCT
ejpam-5587	409	1	this	this	PRON
ejpam-5587	409	2	proves	prove	VERB
ejpam-5587	409	3	the	the	DET
ejpam-5587	409	4	result	result	NOUN
ejpam-5587	409	5	the	the	DET
ejpam-5587	409	6	following	follow	VERB
ejpam-5587	409	7	example	example	NOUN
ejpam-5587	409	8	shows	show	VERB
ejpam-5587	409	9	the	the	DET
ejpam-5587	409	10	following	follow	VERB
ejpam-5587	409	11	facts	fact	NOUN
ejpam-5587	409	12	:	:	PUNCT
ejpam-5587	409	13	•	•	NOUN
ejpam-5587	409	14	theorem	theorem	NOUN
ejpam-5587	409	15	9	9	NUM
ejpam-5587	409	16	does	do	AUX
ejpam-5587	409	17	not	not	PART
ejpam-5587	409	18	hold	hold	VERB
ejpam-5587	409	19	if	if	SCONJ
ejpam-5587	409	20	ker(t	ker(t	NOUN
ejpam-5587	409	21	)	)	PUNCT
ejpam-5587	409	22	̸=	̸=	PROPN
ejpam-5587	409	23	ker(t	ker(t	NOUN
ejpam-5587	409	24	∗	∗	NOUN
ejpam-5587	409	25	)	)	PUNCT
ejpam-5587	409	26	.	.	PUNCT
ejpam-5587	410	1	•	•	INTJ
ejpam-5587	410	2	it	it	PRON
ejpam-5587	410	3	is	be	AUX
ejpam-5587	410	4	well	well	ADV
ejpam-5587	410	5	known	know	VERB
ejpam-5587	410	6	that	that	SCONJ
ejpam-5587	410	7	the	the	DET
ejpam-5587	410	8	residual	residual	ADJ
ejpam-5587	410	9	spectrum	spectrum	NOUN
ejpam-5587	410	10	of	of	ADP
ejpam-5587	410	11	a	a	DET
ejpam-5587	410	12	closed	closed	ADJ
ejpam-5587	410	13	densely	densely	ADV
ejpam-5587	410	14	defined	define	VERB
ejpam-5587	410	15	normal	normal	ADJ
ejpam-5587	410	16	operator	operator	NOUN
ejpam-5587	410	17	is	be	AUX
ejpam-5587	410	18	empty	empty	ADJ
ejpam-5587	410	19	.	.	PUNCT
ejpam-5587	411	1	but	but	CCONJ
ejpam-5587	411	2	this	this	PRON
ejpam-5587	411	3	is	be	AUX
ejpam-5587	411	4	not	not	PART
ejpam-5587	411	5	true	true	ADJ
ejpam-5587	411	6	in	in	ADP
ejpam-5587	411	7	the	the	DET
ejpam-5587	411	8	case	case	NOUN
ejpam-5587	411	9	of	of	ADP
ejpam-5587	411	10	totally	totally	ADV
ejpam-5587	411	11	paranormal	paranormal	ADJ
ejpam-5587	411	12	operators	operator	NOUN
ejpam-5587	411	13	.	.	PUNCT
ejpam-5587	412	1	s.	s.	PROPN
ejpam-5587	412	2	alnabulsi	alnabulsi	PROPN
ejpam-5587	412	3	,	,	PUNCT
ejpam-5587	412	4	m.h.m	m.h.m	PROPN
ejpam-5587	412	5	.	.	PUNCT
ejpam-5587	412	6	rashid	rashid	PROPN
ejpam-5587	412	7	/	/	SYM
ejpam-5587	412	8	eur	eur	PROPN
ejpam-5587	412	9	.	.	PUNCT
ejpam-5587	413	1	j.	j.	PROPN
ejpam-5587	413	2	pure	pure	PROPN
ejpam-5587	413	3	appl	appl	PROPN
ejpam-5587	413	4	.	.	PROPN
ejpam-5587	413	5	math	math	PROPN
ejpam-5587	413	6	,	,	PUNCT
ejpam-5587	413	7	18	18	NUM
ejpam-5587	413	8	(	(	PUNCT
ejpam-5587	413	9	1	1	NUM
ejpam-5587	413	10	)	)	PUNCT
ejpam-5587	413	11	(	(	PUNCT
ejpam-5587	413	12	2025	2025	NUM
ejpam-5587	413	13	)	)	PUNCT
ejpam-5587	413	14	,	,	PUNCT
ejpam-5587	413	15	5587	5587	NUM
ejpam-5587	413	16	12	12	NUM
ejpam-5587	413	17	of	of	ADP
ejpam-5587	413	18	20	20	NUM
ejpam-5587	413	19	example	example	NOUN
ejpam-5587	413	20	3	3	NUM
ejpam-5587	413	21	.	.	PUNCT
ejpam-5587	414	1	let	let	VERB
ejpam-5587	414	2	t	t	NOUN
ejpam-5587	414	3	:	:	PUNCT
ejpam-5587	414	4	ℓ2(n	ℓ2(n	ADJ
ejpam-5587	414	5	)	)	PUNCT
ejpam-5587	414	6	→	→	SYM
ejpam-5587	414	7	ℓ2(n	ℓ2(n	VERB
ejpam-5587	414	8	)	)	PUNCT
ejpam-5587	414	9	be	be	AUX
ejpam-5587	414	10	defined	define	VERB
ejpam-5587	414	11	by	by	ADP
ejpam-5587	414	12	t	t	PROPN
ejpam-5587	414	13	(	(	PUNCT
ejpam-5587	414	14	x1	x1	PROPN
ejpam-5587	414	15	,	,	PUNCT
ejpam-5587	414	16	x2	x2	PROPN
ejpam-5587	414	17	,	,	PUNCT
ejpam-5587	414	18	·	·	PUNCT
ejpam-5587	414	19	·	·	PUNCT
ejpam-5587	414	20	·	·	PUNCT
ejpam-5587	414	21	)	)	PUNCT
ejpam-5587	415	1	=	=	PUNCT
ejpam-5587	415	2	(	(	PUNCT
ejpam-5587	415	3	0	0	NUM
ejpam-5587	415	4	,	,	PUNCT
ejpam-5587	415	5	x1	x1	PROPN
ejpam-5587	415	6	,	,	PUNCT
ejpam-5587	415	7	2x2	2x2	NUM
ejpam-5587	415	8	,	,	PUNCT
ejpam-5587	415	9	3x3	3x3	NUM
ejpam-5587	415	10	,	,	PUNCT
ejpam-5587	415	11	·	·	PUNCT
ejpam-5587	415	12	·	·	PUNCT
ejpam-5587	415	13	·	·	PUNCT
ejpam-5587	415	14	)	)	PUNCT
ejpam-5587	415	15	,	,	PUNCT
ejpam-5587	415	16	where	where	SCONJ
ejpam-5587	415	17	d(t	d(t	PROPN
ejpam-5587	415	18	)	)	PUNCT
ejpam-5587	415	19	=	=	PRON
ejpam-5587	415	20	{	{	PUNCT
ejpam-5587	415	21	(	(	PUNCT
ejpam-5587	415	22	x1	x1	PROPN
ejpam-5587	415	23	,	,	PUNCT
ejpam-5587	415	24	x2	x2	PROPN
ejpam-5587	415	25	,	,	PUNCT
ejpam-5587	415	26	·	·	PUNCT
ejpam-5587	415	27	·	·	PUNCT
ejpam-5587	415	28	·	·	PUNCT
ejpam-5587	415	29	)	)	PUNCT
ejpam-5587	416	1	∈	∈	PROPN
ejpam-5587	416	2	ℓ2(n	ℓ2(n	PROPN
ejpam-5587	416	3	)	)	PUNCT
ejpam-5587	416	4	:	:	PUNCT
ejpam-5587	417	1	∑+∞	∑+∞	ADJ
ejpam-5587	417	2	j=1	j=1	PUNCT
ejpam-5587	417	3	∥jxj∥	∥jxj∥	PROPN
ejpam-5587	417	4	2	2	NUM
ejpam-5587	417	5	<	<	X
ejpam-5587	417	6	+	+	NOUN
ejpam-5587	417	7	∞	∞	NUM
ejpam-5587	417	8	}	}	PUNCT
ejpam-5587	417	9	.	.	PUNCT
ejpam-5587	418	1	as	as	ADP
ejpam-5587	418	2	c0	c0	PROPN
ejpam-5587	418	3	,	,	PUNCT
ejpam-5587	418	4	the	the	DET
ejpam-5587	418	5	space	space	NOUN
ejpam-5587	418	6	of	of	ADP
ejpam-5587	418	7	all	all	DET
ejpam-5587	418	8	complex	complex	ADJ
ejpam-5587	418	9	sequences	sequence	NOUN
ejpam-5587	418	10	consisting	consist	VERB
ejpam-5587	418	11	of	of	ADP
ejpam-5587	418	12	at	at	ADV
ejpam-5587	418	13	most	most	ADV
ejpam-5587	418	14	finitely	finitely	ADV
ejpam-5587	418	15	many	many	ADJ
ejpam-5587	418	16	non	non	ADJ
ejpam-5587	418	17	zero	zero	NUM
ejpam-5587	418	18	terms	term	NOUN
ejpam-5587	418	19	is	be	AUX
ejpam-5587	418	20	a	a	DET
ejpam-5587	418	21	subset	subset	NOUN
ejpam-5587	418	22	of	of	ADP
ejpam-5587	418	23	d(t	d(t	PROPN
ejpam-5587	418	24	)	)	PUNCT
ejpam-5587	418	25	and	and	CCONJ
ejpam-5587	418	26	is	be	AUX
ejpam-5587	418	27	dense	dense	ADJ
ejpam-5587	418	28	in	in	ADP
ejpam-5587	418	29	ℓ2(n	ℓ2(n	PROPN
ejpam-5587	418	30	)	)	PUNCT
ejpam-5587	418	31	,	,	PUNCT
ejpam-5587	418	32	we	we	PRON
ejpam-5587	418	33	can	can	AUX
ejpam-5587	418	34	conclude	conclude	VERB
ejpam-5587	418	35	that	that	PRON
ejpam-5587	418	36	t	t	PROPN
ejpam-5587	418	37	is	be	AUX
ejpam-5587	418	38	densely	densely	ADV
ejpam-5587	418	39	defined	define	VERB
ejpam-5587	418	40	.	.	PUNCT
ejpam-5587	419	1	hence	hence	ADV
ejpam-5587	419	2	t	t	PROPN
ejpam-5587	419	3	∗	∗	NOUN
ejpam-5587	419	4	is	be	AUX
ejpam-5587	419	5	well	well	ADV
ejpam-5587	419	6	defined	define	VERB
ejpam-5587	419	7	.	.	PUNCT
ejpam-5587	420	1	note	note	VERB
ejpam-5587	420	2	that	that	SCONJ
ejpam-5587	420	3	t	t	PROPN
ejpam-5587	420	4	is	be	AUX
ejpam-5587	420	5	a	a	DET
ejpam-5587	420	6	closed	closed	ADJ
ejpam-5587	420	7	operator	operator	NOUN
ejpam-5587	420	8	.	.	PUNCT
ejpam-5587	421	1	we	we	PRON
ejpam-5587	421	2	can	can	AUX
ejpam-5587	421	3	show	show	VERB
ejpam-5587	421	4	that	that	SCONJ
ejpam-5587	421	5	t	t	PROPN
ejpam-5587	421	6	∗(x1	∗(x1	PROPN
ejpam-5587	421	7	,	,	PUNCT
ejpam-5587	421	8	x2	x2	PROPN
ejpam-5587	421	9	,	,	PUNCT
ejpam-5587	421	10	·	·	PUNCT
ejpam-5587	421	11	·	·	PUNCT
ejpam-5587	421	12	·	·	PUNCT
ejpam-5587	421	13	)	)	PUNCT
ejpam-5587	422	1	=	=	PUNCT
ejpam-5587	422	2	(	(	PUNCT
ejpam-5587	422	3	x2	x2	PROPN
ejpam-5587	422	4	,	,	PUNCT
ejpam-5587	422	5	2x3	2x3	NUM
ejpam-5587	422	6	,	,	PUNCT
ejpam-5587	422	7	3x4	3x4	NUM
ejpam-5587	422	8	,	,	PUNCT
ejpam-5587	422	9	·	·	PUNCT
ejpam-5587	422	10	·	·	PUNCT
ejpam-5587	422	11	·	·	PUNCT
ejpam-5587	422	12	)	)	PUNCT
ejpam-5587	422	13	with	with	ADP
ejpam-5587	422	14	d(t	d(t	PROPN
ejpam-5587	422	15	∗	∗	NOUN
ejpam-5587	422	16	)	)	PUNCT
ejpam-5587	423	1	=	=	PRON
ejpam-5587	423	2	{	{	PUNCT
ejpam-5587	423	3	(	(	PUNCT
ejpam-5587	423	4	xn	xn	X
ejpam-5587	423	5	)	)	PUNCT
ejpam-5587	423	6	∈	∈	PROPN
ejpam-5587	423	7	ℓ2(n	ℓ2(n	PROPN
ejpam-5587	423	8	)	)	PUNCT
ejpam-5587	423	9	:	:	PUNCT
ejpam-5587	423	10	∑+∞	∑+∞	ADJ
ejpam-5587	423	11	j=2	j=2	PROPN
ejpam-5587	423	12	∥(j	∥(j	NOUN
ejpam-5587	423	13	−	−	NOUN
ejpam-5587	424	1	1)xj∥2	1)xj∥2	NUM
ejpam-5587	424	2	<	<	X
ejpam-5587	425	1	+	+	NOUN
ejpam-5587	425	2	∞	∞	NUM
ejpam-5587	425	3	}	}	PUNCT
ejpam-5587	425	4	.	.	PUNCT
ejpam-5587	426	1	for	for	ADP
ejpam-5587	426	2	any	any	DET
ejpam-5587	426	3	x	x	SYM
ejpam-5587	426	4	=	=	SYM
ejpam-5587	426	5	(	(	PUNCT
ejpam-5587	426	6	xn	xn	X
ejpam-5587	426	7	)	)	PUNCT
ejpam-5587	426	8	∈	∈	PROPN
ejpam-5587	426	9	d((t	d((t	NOUN
ejpam-5587	426	10	−	−	NOUN
ejpam-5587	426	11	λi)2	λi)2	NOUN
ejpam-5587	426	12	)	)	PUNCT
ejpam-5587	426	13	and	and	CCONJ
ejpam-5587	426	14	λ	λ	X
ejpam-5587	426	15	∈	∈	PROPN
ejpam-5587	426	16	c	c	X
ejpam-5587	426	17	,	,	PUNCT
ejpam-5587	426	18	we	we	PRON
ejpam-5587	426	19	have	have	AUX
ejpam-5587	426	20	∥(t	∥(t	VERB
ejpam-5587	427	1	−	−	ADP
ejpam-5587	427	2	λi)x∥2	λi)x∥2	NOUN
ejpam-5587	428	1	=	=	PUNCT
ejpam-5587	429	1	+	+	ADJ
ejpam-5587	429	2	∞∑	∞∑	NUM
ejpam-5587	429	3	j=1	j=1	ADJ
ejpam-5587	429	4	∥(j	∥(j	NOUN
ejpam-5587	429	5	−	−	NOUN
ejpam-5587	429	6	λ)xj∥2	λ)xj∥2	NUM
ejpam-5587	430	1	≤	≤	NOUN
ejpam-5587	431	1	+	+	PUNCT
ejpam-5587	431	2	∞∑	∞∑	ADJ
ejpam-5587	431	3	j=1	j=1	NOUN
ejpam-5587	431	4	(	(	PUNCT
ejpam-5587	431	5	j	j	PROPN
ejpam-5587	431	6	+	+	PROPN
ejpam-5587	431	7	1−	1−	NUM
ejpam-5587	431	8	λ)(j	λ)(j	PUNCT
ejpam-5587	431	9	−	−	PROPN
ejpam-5587	431	10	λ	λ	SYM
ejpam-5587	431	11	)	)	PUNCT
ejpam-5587	431	12	∥xj∥2	∥xj∥2	X
ejpam-5587	431	13	≤	≤	NOUN
ejpam-5587	431	14	+∞∑	+∞∑	NOUN
ejpam-5587	431	15	j=1	j=1	NOUN
ejpam-5587	431	16	(	(	PUNCT
ejpam-5587	431	17	(	(	PUNCT
ejpam-5587	431	18	j	j	PROPN
ejpam-5587	431	19	+	+	NOUN
ejpam-5587	431	20	1−	1−	NUM
ejpam-5587	431	21	λ)(j	λ)(j	PUNCT
ejpam-5587	431	22	−	−	PROPN
ejpam-5587	431	23	λ	λ	PROPN
ejpam-5587	431	24	)	)	PUNCT
ejpam-5587	431	25	∥xj∥)2	∥xj∥)2	NOUN
ejpam-5587	431	26			PROPN
ejpam-5587	431	27	1	1	NUM
ejpam-5587	431	28	2	2	NUM
ejpam-5587	431	29	+∞∑	+∞∑	NOUN
ejpam-5587	431	30	j=1	j=1	PROPN
ejpam-5587	431	31	∥xj∥2	∥xj∥2	PUNCT
ejpam-5587	432	1			PROPN
ejpam-5587	432	2	1	1	NUM
ejpam-5587	432	3	2	2	NUM
ejpam-5587	432	4	≤	≤	NOUN
ejpam-5587	432	5	∥∥(t	∥∥(t	PUNCT
ejpam-5587	432	6	−	−	NOUN
ejpam-5587	432	7	λi)2x	λi)2x	VERB
ejpam-5587	432	8	∥∥	∥∥	X
ejpam-5587	432	9	∥x∥	∥x∥	NOUN
ejpam-5587	432	10	.	.	PUNCT
ejpam-5587	433	1	hence	hence	ADV
ejpam-5587	433	2	t	t	PROPN
ejpam-5587	433	3	is	be	AUX
ejpam-5587	433	4	totally	totally	ADV
ejpam-5587	433	5	paranormal	paranormal	ADJ
ejpam-5587	433	6	.	.	PUNCT
ejpam-5587	434	1	since	since	SCONJ
ejpam-5587	434	2	∥tx∥	∥tx∥	PROPN
ejpam-5587	434	3	≥	≥	NUM
ejpam-5587	434	4	∥x∥	∥x∥	NOUN
ejpam-5587	434	5	for	for	ADP
ejpam-5587	434	6	all	all	DET
ejpam-5587	434	7	x	x	SYM
ejpam-5587	434	8	∈	∈	PROPN
ejpam-5587	434	9	d(t	d(t	PROPN
ejpam-5587	434	10	)	)	PUNCT
ejpam-5587	434	11	and	and	CCONJ
ejpam-5587	434	12	∥te1∥	∥te1∥	NOUN
ejpam-5587	434	13	=	=	SYM
ejpam-5587	434	14	∥e1∥	∥e1∥	PROPN
ejpam-5587	434	15	,	,	PUNCT
ejpam-5587	434	16	we	we	PRON
ejpam-5587	434	17	get	get	VERB
ejpam-5587	434	18	m(t	m(t	NOUN
ejpam-5587	434	19	)	)	PUNCT
ejpam-5587	435	1	=	=	PUNCT
ejpam-5587	435	2	1	1	X
ejpam-5587	435	3	.	.	X
ejpam-5587	435	4	also	also	ADV
ejpam-5587	435	5	it	it	PRON
ejpam-5587	435	6	can	can	AUX
ejpam-5587	435	7	be	be	AUX
ejpam-5587	435	8	easily	easily	ADV
ejpam-5587	435	9	verified	verify	VERB
ejpam-5587	435	10	that	that	SCONJ
ejpam-5587	435	11	t	t	PROPN
ejpam-5587	435	12	is	be	AUX
ejpam-5587	435	13	injective	injective	ADJ
ejpam-5587	435	14	,	,	PUNCT
ejpam-5587	435	15	ran(t	ran(t	X
ejpam-5587	435	16	)	)	PUNCT
ejpam-5587	435	17	=	=	PUNCT
ejpam-5587	435	18	ℓ2(n	ℓ2(n	ADJ
ejpam-5587	435	19	)	)	PUNCT
ejpam-5587	435	20	\	\	NOUN
ejpam-5587	435	21	span	span	NOUN
ejpam-5587	435	22	{	{	PUNCT
ejpam-5587	435	23	e1	e1	PROPN
ejpam-5587	435	24	}	}	PUNCT
ejpam-5587	435	25	is	be	AUX
ejpam-5587	435	26	closed	close	VERB
ejpam-5587	435	27	but	but	CCONJ
ejpam-5587	435	28	ran(t	ran(t	NUM
ejpam-5587	435	29	)	)	PUNCT
ejpam-5587	436	1	̸=	̸=	PROPN
ejpam-5587	436	2	h	h	NOUN
ejpam-5587	436	3	,	,	PUNCT
ejpam-5587	436	4	so	so	ADV
ejpam-5587	436	5	0	0	NUM
ejpam-5587	436	6	∈	∈	PROPN
ejpam-5587	436	7	σ(t	σ(t	PROPN
ejpam-5587	436	8	)	)	PUNCT
ejpam-5587	436	9	.	.	PUNCT
ejpam-5587	437	1	hence	hence	ADV
ejpam-5587	437	2	d(0	d(0	PROPN
ejpam-5587	437	3	,	,	PUNCT
ejpam-5587	437	4	σ(t	σ(t	PROPN
ejpam-5587	437	5	)	)	PUNCT
ejpam-5587	437	6	)	)	PUNCT
ejpam-5587	438	1	=	=	SYM
ejpam-5587	438	2	0	0	NUM
ejpam-5587	439	1	̸=	̸=	PROPN
ejpam-5587	439	2	1	1	NUM
ejpam-5587	439	3	=	=	SYM
ejpam-5587	439	4	m(t	m(t	PROPN
ejpam-5587	439	5	)	)	PUNCT
ejpam-5587	439	6	.	.	PUNCT
ejpam-5587	440	1	now	now	ADV
ejpam-5587	440	2	we	we	PRON
ejpam-5587	440	3	will	will	AUX
ejpam-5587	440	4	show	show	VERB
ejpam-5587	440	5	that	that	SCONJ
ejpam-5587	440	6	σ(t	σ(t	PROPN
ejpam-5587	440	7	)	)	PUNCT
ejpam-5587	441	1	=	=	SYM
ejpam-5587	441	2	c.	c.	NOUN
ejpam-5587	441	3	to	to	PART
ejpam-5587	441	4	prove	prove	VERB
ejpam-5587	441	5	this	this	PRON
ejpam-5587	441	6	,	,	PUNCT
ejpam-5587	441	7	we	we	PRON
ejpam-5587	441	8	show	show	VERB
ejpam-5587	441	9	that	that	SCONJ
ejpam-5587	441	10	t	t	PROPN
ejpam-5587	441	11	−	−	PROPN
ejpam-5587	442	1	µi	µi	PROPN
ejpam-5587	442	2	is	be	AUX
ejpam-5587	442	3	injective	injective	ADJ
ejpam-5587	442	4	and	and	CCONJ
ejpam-5587	442	5	ker(t	ker(t	NOUN
ejpam-5587	442	6	−	−	NOUN
ejpam-5587	442	7	µi)∗	µi)∗	X
ejpam-5587	442	8	̸=	̸=	PROPN
ejpam-5587	442	9	{	{	PUNCT
ejpam-5587	442	10	0	0	NUM
ejpam-5587	442	11	}	}	PUNCT
ejpam-5587	442	12	,	,	PUNCT
ejpam-5587	442	13	for	for	SCONJ
ejpam-5587	442	14	all	all	DET
ejpam-5587	442	15	µ	µ	PRON
ejpam-5587	442	16	∈	∈	PROPN
ejpam-5587	442	17	c.	c.	NOUN
ejpam-5587	442	18	let	let	VERB
ejpam-5587	442	19	µ	µ	X
ejpam-5587	442	20	∈	∈	X
ejpam-5587	442	21	c	c	NOUN
ejpam-5587	442	22	\	\	X
ejpam-5587	442	23	{	{	PUNCT
ejpam-5587	442	24	0	0	NUM
ejpam-5587	442	25	}	}	PUNCT
ejpam-5587	442	26	and	and	CCONJ
ejpam-5587	442	27	(	(	PUNCT
ejpam-5587	442	28	t	t	PROPN
ejpam-5587	442	29	−	−	PROPN
ejpam-5587	442	30	µi))x	µi))x	PROPN
ejpam-5587	442	31	=	=	PROPN
ejpam-5587	442	32	0	0	PROPN
ejpam-5587	442	33	for	for	ADP
ejpam-5587	442	34	some	some	PRON
ejpam-5587	442	35	x	x	X
ejpam-5587	442	36	=	=	SYM
ejpam-5587	442	37	(	(	PUNCT
ejpam-5587	442	38	xn	xn	X
ejpam-5587	442	39	)	)	PUNCT
ejpam-5587	442	40	∈	∈	PROPN
ejpam-5587	442	41	d(t	d(t	PROPN
ejpam-5587	442	42	)	)	PUNCT
ejpam-5587	442	43	.	.	PUNCT
ejpam-5587	443	1	then	then	ADV
ejpam-5587	443	2	(	(	PUNCT
ejpam-5587	443	3	−µx1	−µx1	X
ejpam-5587	443	4	,	,	PUNCT
ejpam-5587	443	5	x1	x1	PROPN
ejpam-5587	443	6	−	−	PROPN
ejpam-5587	443	7	µx2	µx2	PROPN
ejpam-5587	443	8	,	,	PUNCT
ejpam-5587	443	9	2x2	2x2	NUM
ejpam-5587	443	10	−	−	NOUN
ejpam-5587	443	11	µx3	µx3	PROPN
ejpam-5587	443	12	,	,	PUNCT
ejpam-5587	443	13	·	·	PUNCT
ejpam-5587	443	14	·	·	PUNCT
ejpam-5587	443	15	·	·	PUNCT
ejpam-5587	443	16	)	)	PUNCT
ejpam-5587	444	1	=	=	PUNCT
ejpam-5587	444	2	0	0	X
ejpam-5587	444	3	.	.	X
ejpam-5587	445	1	equating	equate	VERB
ejpam-5587	445	2	component	component	NOUN
ejpam-5587	445	3	-	-	PUNCT
ejpam-5587	445	4	wise	wise	ADJ
ejpam-5587	445	5	we	we	PRON
ejpam-5587	445	6	get	get	VERB
ejpam-5587	445	7	x	x	X
ejpam-5587	445	8	=	=	SYM
ejpam-5587	445	9	0	0	NUM
ejpam-5587	445	10	.	.	PUNCT
ejpam-5587	446	1	this	this	PRON
ejpam-5587	446	2	implies	imply	VERB
ejpam-5587	446	3	that	that	SCONJ
ejpam-5587	446	4	t	t	PROPN
ejpam-5587	446	5	−µi	−µi	NOUN
ejpam-5587	446	6	injective	injective	ADJ
ejpam-5587	446	7	.	.	PUNCT
ejpam-5587	447	1	let	let	VERB
ejpam-5587	447	2	y	y	PROPN
ejpam-5587	447	3	=	=	SYM
ejpam-5587	447	4	(	(	PUNCT
ejpam-5587	447	5	yn	yn	NOUN
ejpam-5587	447	6	)	)	PUNCT
ejpam-5587	447	7	∈	∈	PROPN
ejpam-5587	447	8	d(t	d(t	PROPN
ejpam-5587	447	9	∗	∗	NOUN
ejpam-5587	447	10	)	)	PUNCT
ejpam-5587	447	11	be	be	VERB
ejpam-5587	447	12	such	such	ADJ
ejpam-5587	447	13	that	that	SCONJ
ejpam-5587	447	14	(	(	PUNCT
ejpam-5587	447	15	t	t	NOUN
ejpam-5587	447	16	−	−	NOUN
ejpam-5587	447	17	µi)∗y	µi)∗y	NUM
ejpam-5587	447	18	=	=	SYM
ejpam-5587	447	19	0	0	NUM
ejpam-5587	447	20	.	.	PUNCT
ejpam-5587	448	1	that	that	PRON
ejpam-5587	448	2	is	be	AUX
ejpam-5587	448	3	,	,	PUNCT
ejpam-5587	448	4	(	(	PUNCT
ejpam-5587	448	5	y2	y2	INTJ
ejpam-5587	448	6	−	−	PROPN
ejpam-5587	448	7	µ̄y1	µ̄y1	NOUN
ejpam-5587	448	8	,	,	PUNCT
ejpam-5587	448	9	2y3	2y3	NUM
ejpam-5587	448	10	−	−	PROPN
ejpam-5587	448	11	µ̄y2	µ̄y2	NOUN
ejpam-5587	448	12	,	,	PUNCT
ejpam-5587	448	13	y4	y4	PROPN
ejpam-5587	448	14	−	−	PROPN
ejpam-5587	448	15	µ̄y3	µ̄y3	NOUN
ejpam-5587	448	16	,	,	PUNCT
ejpam-5587	448	17	·	·	PUNCT
ejpam-5587	448	18	·	·	PUNCT
ejpam-5587	448	19	·	·	PUNCT
ejpam-5587	448	20	)	)	PUNCT
ejpam-5587	449	1	=	=	PUNCT
ejpam-5587	449	2	0	0	X
ejpam-5587	449	3	.	.	PUNCT
ejpam-5587	450	1	from	from	ADP
ejpam-5587	450	2	this	this	PRON
ejpam-5587	450	3	we	we	PRON
ejpam-5587	450	4	get	get	VERB
ejpam-5587	450	5	y	y	NOUN
ejpam-5587	450	6	=	=	PUNCT
ejpam-5587	450	7	(	(	PUNCT
ejpam-5587	450	8	1	1	NUM
ejpam-5587	450	9	,	,	PUNCT
ejpam-5587	450	10	µ̄	µ̄	PROPN
ejpam-5587	450	11	,	,	PUNCT
ejpam-5587	450	12	(	(	PUNCT
ejpam-5587	450	13	µ̄)2	µ̄)2	PROPN
ejpam-5587	450	14	2	2	NUM
ejpam-5587	450	15	!	!	NUM
ejpam-5587	450	16	,	,	PUNCT
ejpam-5587	450	17	(	(	PUNCT
ejpam-5587	450	18	µ̄)3	µ̄)3	PROPN
ejpam-5587	450	19	3	3	NUM
ejpam-5587	450	20	!	!	NUM
ejpam-5587	450	21	,	,	PUNCT
ejpam-5587	450	22	·	·	PUNCT
ejpam-5587	451	1	·	·	PUNCT
ejpam-5587	451	2	·	·	PUNCT
ejpam-5587	451	3	)	)	PUNCT
ejpam-5587	451	4	y1	y1	INTJ
ejpam-5587	451	5	.	.	PUNCT
ejpam-5587	452	1	(	(	PUNCT
ejpam-5587	452	2	3	3	X
ejpam-5587	452	3	)	)	PUNCT
ejpam-5587	452	4	if	if	SCONJ
ejpam-5587	452	5	µ	µ	X
ejpam-5587	452	6	=	=	SYM
ejpam-5587	452	7	0	0	NUM
ejpam-5587	452	8	,	,	PUNCT
ejpam-5587	452	9	then	then	ADV
ejpam-5587	452	10	ker(t	ker(t	NOUN
ejpam-5587	452	11	∗	∗	NOUN
ejpam-5587	452	12	)	)	PUNCT
ejpam-5587	452	13	=	=	SYM
ejpam-5587	452	14	span{e1	span{e1	NOUN
ejpam-5587	452	15	}	}	PUNCT
ejpam-5587	452	16	.	.	PUNCT
ejpam-5587	453	1	if	if	SCONJ
ejpam-5587	453	2	µ	µ	DET
ejpam-5587	453	3	̸=	̸=	PROPN
ejpam-5587	453	4	0	0	NUM
ejpam-5587	453	5	,	,	PUNCT
ejpam-5587	453	6	then	then	ADV
ejpam-5587	453	7	we	we	PRON
ejpam-5587	453	8	will	will	AUX
ejpam-5587	453	9	show	show	VERB
ejpam-5587	453	10	that	that	SCONJ
ejpam-5587	453	11	y	y	PROPN
ejpam-5587	453	12	obtained	obtain	VERB
ejpam-5587	453	13	in	in	ADP
ejpam-5587	453	14	equation	equation	NOUN
ejpam-5587	453	15	(	(	PUNCT
ejpam-5587	453	16	3	3	X
ejpam-5587	453	17	)	)	PUNCT
ejpam-5587	453	18	belongs	belong	VERB
ejpam-5587	453	19	to	to	ADP
ejpam-5587	453	20	ker(t	ker(t	NOUN
ejpam-5587	453	21	−	−	NOUN
ejpam-5587	453	22	µi)∗.	µi)∗.	ADV
ejpam-5587	453	23	consider	consider	VERB
ejpam-5587	453	24	zn	zn	NOUN
ejpam-5587	453	25	=	=	SYM
ejpam-5587	453	26	µ̄2n	µ̄2n	X
ejpam-5587	453	27	(	(	PUNCT
ejpam-5587	453	28	n!)2	n!)2	ADV
ejpam-5587	453	29	.	.	PUNCT
ejpam-5587	454	1	then∣∣∣∣zn+1	then∣∣∣∣zn+1	NOUN
ejpam-5587	454	2	zn	zn	X
ejpam-5587	454	3	∣∣∣∣	∣∣∣∣	NOUN
ejpam-5587	454	4	=	=	SYM
ejpam-5587	454	5	|µ|2	|µ|2	ADJ
ejpam-5587	454	6	n+	n+	PUNCT
ejpam-5587	454	7	1	1	NUM
ejpam-5587	454	8	→	→	SYM
ejpam-5587	454	9	0	0	NUM
ejpam-5587	454	10	as	as	ADP
ejpam-5587	454	11	n	n	PRON
ejpam-5587	454	12	→	→	PUNCT
ejpam-5587	454	13	+	+	PROPN
ejpam-5587	454	14	∞.	∞.	PROPN
ejpam-5587	454	15	s.	s.	PROPN
ejpam-5587	454	16	alnabulsi	alnabulsi	PROPN
ejpam-5587	454	17	,	,	PUNCT
ejpam-5587	454	18	m.h.m	m.h.m	PROPN
ejpam-5587	454	19	.	.	PUNCT
ejpam-5587	454	20	rashid	rashid	PROPN
ejpam-5587	454	21	/	/	SYM
ejpam-5587	454	22	eur	eur	PROPN
ejpam-5587	454	23	.	.	PUNCT
ejpam-5587	455	1	j.	j.	PROPN
ejpam-5587	455	2	pure	pure	PROPN
ejpam-5587	455	3	appl	appl	PROPN
ejpam-5587	455	4	.	.	PROPN
ejpam-5587	455	5	math	math	PROPN
ejpam-5587	455	6	,	,	PUNCT
ejpam-5587	455	7	18	18	NUM
ejpam-5587	455	8	(	(	PUNCT
ejpam-5587	455	9	1	1	NUM
ejpam-5587	455	10	)	)	PUNCT
ejpam-5587	455	11	(	(	PUNCT
ejpam-5587	455	12	2025	2025	NUM
ejpam-5587	455	13	)	)	PUNCT
ejpam-5587	455	14	,	,	PUNCT
ejpam-5587	455	15	5587	5587	NUM
ejpam-5587	455	16	13	13	NUM
ejpam-5587	455	17	of	of	ADP
ejpam-5587	455	18	20	20	NUM
ejpam-5587	455	19	by	by	ADP
ejpam-5587	455	20	the	the	DET
ejpam-5587	455	21	ratio	ratio	NOUN
ejpam-5587	455	22	test	test	NOUN
ejpam-5587	455	23	we	we	PRON
ejpam-5587	455	24	conclude	conclude	VERB
ejpam-5587	455	25	that	that	PRON
ejpam-5587	455	26	∑+∞	∑+∞	PUNCT
ejpam-5587	455	27	j=1	j=1	PROPN
ejpam-5587	455	28	zn	zn	PROPN
ejpam-5587	455	29	is	be	AUX
ejpam-5587	455	30	absolutely	absolutely	ADV
ejpam-5587	455	31	convergent	convergent	ADJ
ejpam-5587	455	32	,	,	PUNCT
ejpam-5587	455	33	that	that	PRON
ejpam-5587	455	34	is	be	AUX
ejpam-5587	455	35	∑+∞	∑+∞	ADJ
ejpam-5587	455	36	n=1	n=1	PROPN
ejpam-5587	455	37	(	(	PUNCT
ejpam-5587	455	38	|µ|n	|µ|n	PROPN
ejpam-5587	455	39	n	n	CCONJ
ejpam-5587	455	40	!	!	PUNCT
ejpam-5587	455	41	)	)	PUNCT
ejpam-5587	456	1	2	2	X
ejpam-5587	456	2	<	<	X
ejpam-5587	456	3	+	+	X
ejpam-5587	456	4	∞.	∞.	PROPN
ejpam-5587	456	5	thus	thus	ADV
ejpam-5587	456	6	y	y	PROPN
ejpam-5587	456	7	∈	∈	PROPN
ejpam-5587	456	8	ℓ2(n	ℓ2(n	PROPN
ejpam-5587	456	9	)	)	PUNCT
ejpam-5587	456	10	.	.	PUNCT
ejpam-5587	457	1	on	on	ADP
ejpam-5587	457	2	the	the	DET
ejpam-5587	457	3	similar	similar	ADJ
ejpam-5587	457	4	lines	line	NOUN
ejpam-5587	457	5	we	we	PRON
ejpam-5587	457	6	can	can	AUX
ejpam-5587	457	7	show	show	VERB
ejpam-5587	457	8	that	that	SCONJ
ejpam-5587	457	9	∑+∞	∑+∞	ADJ
ejpam-5587	457	10	n=1	n=1	PROPN
ejpam-5587	457	11	(	(	PUNCT
ejpam-5587	457	12	|µ|n	|µ|n	X
ejpam-5587	457	13	(	(	PUNCT
ejpam-5587	457	14	n−1	n−1	PROPN
ejpam-5587	457	15	)	)	PUNCT
ejpam-5587	457	16	!	!	PUNCT
ejpam-5587	457	17	)	)	PUNCT
ejpam-5587	458	1	2	2	X
ejpam-5587	458	2	<	<	X
ejpam-5587	458	3	+	+	NOUN
ejpam-5587	458	4	∞.	∞.	PROPN
ejpam-5587	458	5	hence	hence	ADV
ejpam-5587	458	6	ker(t	ker(t	PROPN
ejpam-5587	458	7	−	−	PROPN
ejpam-5587	458	8	µi))∗	µi))∗	NOUN
ejpam-5587	458	9	̸=	̸=	PROPN
ejpam-5587	458	10	{	{	PUNCT
ejpam-5587	458	11	0	0	NUM
ejpam-5587	458	12	}	}	PUNCT
ejpam-5587	458	13	.	.	PUNCT
ejpam-5587	459	1	for	for	ADP
ejpam-5587	459	2	every	every	DET
ejpam-5587	459	3	µ	µ	PROPN
ejpam-5587	459	4	∈	∈	ADJ
ejpam-5587	459	5	c	c	NOUN
ejpam-5587	459	6	,	,	PUNCT
ejpam-5587	459	7	ker(t	ker(t	NOUN
ejpam-5587	459	8	−	−	PROPN
ejpam-5587	459	9	µi	µi	PROPN
ejpam-5587	459	10	)	)	PUNCT
ejpam-5587	459	11	=	=	PUNCT
ejpam-5587	459	12	{	{	PUNCT
ejpam-5587	459	13	0	0	NUM
ejpam-5587	459	14	}	}	PUNCT
ejpam-5587	459	15	and	and	CCONJ
ejpam-5587	459	16	ran(t	ran(t	NUM
ejpam-5587	459	17	−	−	PROPN
ejpam-5587	459	18	µi	µi	PROPN
ejpam-5587	459	19	)	)	PUNCT
ejpam-5587	459	20	=	=	SYM
ejpam-5587	459	21	(	(	PUNCT
ejpam-5587	459	22	ker(t	ker(t	NOUN
ejpam-5587	459	23	−	−	NOUN
ejpam-5587	459	24	µi)∗)⊥	µi)∗)⊥	ADJ
ejpam-5587	459	25	̸=	̸=	PROPN
ejpam-5587	459	26	ℓ2(n	ℓ2(n	PROPN
ejpam-5587	459	27	)	)	PUNCT
ejpam-5587	459	28	.	.	PUNCT
ejpam-5587	460	1	hence	hence	ADV
ejpam-5587	460	2	we	we	PRON
ejpam-5587	460	3	conclude	conclude	VERB
ejpam-5587	460	4	that	that	SCONJ
ejpam-5587	460	5	µ	µ	X
ejpam-5587	460	6	∈	∈	PROPN
ejpam-5587	460	7	σr(t	σr(t	NOUN
ejpam-5587	460	8	)	)	PUNCT
ejpam-5587	460	9	,	,	PUNCT
ejpam-5587	460	10	and	and	CCONJ
ejpam-5587	460	11	σ(t	σ(t	PROPN
ejpam-5587	460	12	)	)	PUNCT
ejpam-5587	461	1	=	=	SYM
ejpam-5587	461	2	c.	c.	NOUN
ejpam-5587	461	3	we	we	PRON
ejpam-5587	461	4	also	also	ADV
ejpam-5587	461	5	have	have	VERB
ejpam-5587	461	6	γ(t	γ(t	NOUN
ejpam-5587	461	7	)	)	PUNCT
ejpam-5587	462	1	=	=	SYM
ejpam-5587	463	1	1	1	NUM
ejpam-5587	463	2	̸=	̸=	PROPN
ejpam-5587	463	3	0	0	NUM
ejpam-5587	463	4	=	=	SYM
ejpam-5587	463	5	d(t	d(t	PROPN
ejpam-5587	463	6	)	)	PUNCT
ejpam-5587	463	7	.	.	PUNCT
ejpam-5587	464	1	from	from	ADP
ejpam-5587	464	2	this	this	PRON
ejpam-5587	464	3	we	we	PRON
ejpam-5587	464	4	conclude	conclude	VERB
ejpam-5587	464	5	that	that	PRON
ejpam-5587	464	6	corollary	corollary	ADJ
ejpam-5587	464	7	2	2	NUM
ejpam-5587	464	8	is	be	AUX
ejpam-5587	464	9	also	also	ADV
ejpam-5587	464	10	not	not	PART
ejpam-5587	464	11	true	true	ADJ
ejpam-5587	464	12	if	if	SCONJ
ejpam-5587	464	13	the	the	DET
ejpam-5587	464	14	condition	condition	NOUN
ejpam-5587	464	15	,	,	PUNCT
ejpam-5587	464	16	ker(t	ker(t	NOUN
ejpam-5587	464	17	)	)	PUNCT
ejpam-5587	464	18	=	=	SYM
ejpam-5587	465	1	ker(t	ker(t	NOUN
ejpam-5587	465	2	∗	∗	NOUN
ejpam-5587	465	3	)	)	PUNCT
ejpam-5587	465	4	is	be	AUX
ejpam-5587	465	5	dropped	drop	VERB
ejpam-5587	465	6	.	.	PUNCT
ejpam-5587	466	1	theorem	theorem	ADJ
ejpam-5587	466	2	10	10	NUM
ejpam-5587	466	3	.	.	PUNCT
ejpam-5587	467	1	suppose	suppose	VERB
ejpam-5587	467	2	t	t	PROPN
ejpam-5587	467	3	∈	∈	PROPN
ejpam-5587	467	4	l(h	l(h	PROPN
ejpam-5587	467	5	)	)	PUNCT
ejpam-5587	467	6	is	be	AUX
ejpam-5587	467	7	a	a	DET
ejpam-5587	467	8	densely	densely	ADV
ejpam-5587	467	9	defined	define	VERB
ejpam-5587	467	10	closed	close	VERB
ejpam-5587	467	11	totally	totally	ADV
ejpam-5587	467	12	paranormal	paranormal	ADJ
ejpam-5587	467	13	operator	operator	NOUN
ejpam-5587	467	14	,	,	PUNCT
ejpam-5587	467	15	ker(t	ker(t	NOUN
ejpam-5587	467	16	)	)	PUNCT
ejpam-5587	468	1	=	=	SYM
ejpam-5587	468	2	ker(t	ker(t	NOUN
ejpam-5587	468	3	∗	∗	NOUN
ejpam-5587	468	4	)	)	PUNCT
ejpam-5587	468	5	and	and	CCONJ
ejpam-5587	468	6	0	0	NUM
ejpam-5587	468	7	is	be	AUX
ejpam-5587	468	8	an	an	DET
ejpam-5587	468	9	isolated	isolated	ADJ
ejpam-5587	468	10	spectral	spectral	ADJ
ejpam-5587	468	11	value	value	NOUN
ejpam-5587	468	12	of	of	ADP
ejpam-5587	468	13	t	t	PROPN
ejpam-5587	468	14	.	.	PUNCT
ejpam-5587	469	1	then	then	ADV
ejpam-5587	469	2	0	0	NUM
ejpam-5587	469	3	∈	∈	NOUN
ejpam-5587	469	4	σp(t	σp(t	PUNCT
ejpam-5587	469	5	)	)	PUNCT
ejpam-5587	469	6	.	.	PUNCT
ejpam-5587	470	1	proof	proof	NOUN
ejpam-5587	470	2	.	.	PUNCT
ejpam-5587	471	1	since	since	SCONJ
ejpam-5587	471	2	0	0	NUM
ejpam-5587	471	3	is	be	AUX
ejpam-5587	471	4	an	an	DET
ejpam-5587	471	5	isolated	isolated	ADJ
ejpam-5587	471	6	spectral	spectral	ADJ
ejpam-5587	471	7	value	value	NOUN
ejpam-5587	471	8	of	of	ADP
ejpam-5587	471	9	t	t	PROPN
ejpam-5587	471	10	,	,	PUNCT
ejpam-5587	471	11	d(t	d(t	PROPN
ejpam-5587	471	12	)	)	PUNCT
ejpam-5587	471	13	>	>	X
ejpam-5587	472	1	0	0	X
ejpam-5587	472	2	.	.	PUNCT
ejpam-5587	472	3	hence	hence	ADV
ejpam-5587	472	4	by	by	ADP
ejpam-5587	472	5	corollary	corollary	ADJ
ejpam-5587	472	6	2	2	NUM
ejpam-5587	472	7	,	,	PUNCT
ejpam-5587	472	8	γ(t	γ(t	PROPN
ejpam-5587	472	9	)	)	PUNCT
ejpam-5587	472	10	>	>	PUNCT
ejpam-5587	472	11	)	)	PUNCT
ejpam-5587	472	12	so	so	SCONJ
ejpam-5587	472	13	that	that	SCONJ
ejpam-5587	472	14	by	by	ADP
ejpam-5587	472	15	theorem	theorem	NOUN
ejpam-5587	472	16	8	8	NUM
ejpam-5587	472	17	,	,	PUNCT
ejpam-5587	472	18	ran(t	ran(t	PROPN
ejpam-5587	472	19	)	)	PUNCT
ejpam-5587	472	20	is	be	AUX
ejpam-5587	472	21	closed	closed	ADJ
ejpam-5587	472	22	.	.	PUNCT
ejpam-5587	473	1	if	if	SCONJ
ejpam-5587	473	2	0	0	NUM
ejpam-5587	473	3	/∈	/∈	NUM
ejpam-5587	473	4	σp(t	σp(t	PUNCT
ejpam-5587	473	5	)	)	PUNCT
ejpam-5587	473	6	,	,	PUNCT
ejpam-5587	473	7	then	then	ADV
ejpam-5587	473	8	ker(t	ker(t	NOUN
ejpam-5587	473	9	)	)	PUNCT
ejpam-5587	474	1	=	=	PUNCT
ejpam-5587	474	2	{	{	PUNCT
ejpam-5587	474	3	0	0	NUM
ejpam-5587	474	4	}	}	PUNCT
ejpam-5587	474	5	so	so	SCONJ
ejpam-5587	474	6	that	that	SCONJ
ejpam-5587	474	7	we	we	PRON
ejpam-5587	474	8	also	also	ADV
ejpam-5587	474	9	have	have	VERB
ejpam-5587	474	10	ran(t	ran(t	NOUN
ejpam-5587	474	11	)	)	PUNCT
ejpam-5587	475	1	=	=	SYM
ejpam-5587	475	2	ran(t	ran(t	NOUN
ejpam-5587	475	3	)	)	PUNCT
ejpam-5587	476	1	=	=	PUNCT
ejpam-5587	476	2	ker(t	ker(t	NOUN
ejpam-5587	476	3	)	)	PUNCT
ejpam-5587	476	4	⊥	⊥	NOUN
ejpam-5587	476	5	=	=	SYM
ejpam-5587	476	6	h	h	NOUN
ejpam-5587	476	7	,	,	PUNCT
ejpam-5587	476	8	making	make	VERB
ejpam-5587	476	9	t	t	NOUN
ejpam-5587	476	10	bijective	bijective	VERB
ejpam-5587	476	11	and	and	CCONJ
ejpam-5587	476	12	hence	hence	ADV
ejpam-5587	476	13	0	0	NUM
ejpam-5587	476	14	/∈	/∈	PUNCT
ejpam-5587	476	15	σ(t	σ(t	PROPN
ejpam-5587	476	16	)	)	PUNCT
ejpam-5587	476	17	,	,	PUNCT
ejpam-5587	476	18	a	a	DET
ejpam-5587	476	19	contradiction	contradiction	NOUN
ejpam-5587	476	20	.	.	PUNCT
ejpam-5587	477	1	hence	hence	ADV
ejpam-5587	477	2	0	0	NUM
ejpam-5587	477	3	∈	∈	NOUN
ejpam-5587	477	4	σp(t	σp(t	PUNCT
ejpam-5587	477	5	)	)	PUNCT
ejpam-5587	477	6	.	.	PUNCT
ejpam-5587	478	1	example	example	NOUN
ejpam-5587	479	1	4	4	NUM
ejpam-5587	479	2	.	.	PUNCT
ejpam-5587	479	3	the	the	DET
ejpam-5587	479	4	converse	converse	NOUN
ejpam-5587	479	5	of	of	ADP
ejpam-5587	479	6	theorem	theorem	NOUN
ejpam-5587	479	7	10	10	NUM
ejpam-5587	479	8	need	need	AUX
ejpam-5587	479	9	not	not	PART
ejpam-5587	479	10	be	be	AUX
ejpam-5587	479	11	true	true	ADJ
ejpam-5587	479	12	.	.	PUNCT
ejpam-5587	480	1	to	to	PART
ejpam-5587	480	2	see	see	VERB
ejpam-5587	480	3	this	this	PRON
ejpam-5587	480	4	,	,	PUNCT
ejpam-5587	480	5	consider	consider	VERB
ejpam-5587	480	6	t	t	NOUN
ejpam-5587	480	7	:	:	PUNCT
ejpam-5587	480	8	ℓ2(n	ℓ2(n	ADJ
ejpam-5587	480	9	)	)	PUNCT
ejpam-5587	480	10	→	→	SYM
ejpam-5587	480	11	ℓ2(n	ℓ2(n	ADJ
ejpam-5587	480	12	)	)	PUNCT
ejpam-5587	480	13	defined	define	VERB
ejpam-5587	480	14	by	by	ADP
ejpam-5587	480	15	t	t	PROPN
ejpam-5587	480	16	(	(	PUNCT
ejpam-5587	480	17	x1	x1	PROPN
ejpam-5587	480	18	,	,	PUNCT
ejpam-5587	480	19	x2	x2	PROPN
ejpam-5587	480	20	,	,	PUNCT
ejpam-5587	480	21	x3	x3	ADJ
ejpam-5587	480	22	,	,	PUNCT
ejpam-5587	480	23	·	·	PUNCT
ejpam-5587	480	24	·	·	PUNCT
ejpam-5587	480	25	·	·	PUNCT
ejpam-5587	480	26	)	)	PUNCT
ejpam-5587	481	1	=	=	PUNCT
ejpam-5587	481	2	(	(	PUNCT
ejpam-5587	481	3	0	0	NUM
ejpam-5587	481	4	,	,	PUNCT
ejpam-5587	481	5	2x2	2x2	NUM
ejpam-5587	481	6	,	,	PUNCT
ejpam-5587	481	7	1	1	NUM
ejpam-5587	481	8	3	3	NUM
ejpam-5587	481	9	x3	x3	ADJ
ejpam-5587	481	10	,	,	PUNCT
ejpam-5587	481	11	4x4	4x4	NUM
ejpam-5587	481	12	,	,	PUNCT
ejpam-5587	481	13	1	1	NUM
ejpam-5587	481	14	5	5	NUM
ejpam-5587	481	15	x5	x5	NOUN
ejpam-5587	481	16	,	,	PUNCT
ejpam-5587	481	17	·	·	PUNCT
ejpam-5587	481	18	·	·	PUNCT
ejpam-5587	481	19	·	·	PUNCT
ejpam-5587	481	20	)	)	PUNCT
ejpam-5587	481	21	,	,	PUNCT
ejpam-5587	481	22	where	where	SCONJ
ejpam-5587	481	23	d(t	d(t	NOUN
ejpam-5587	481	24	)	)	PUNCT
ejpam-5587	481	25	=	=	PRON
ejpam-5587	482	1	{	{	PUNCT
ejpam-5587	482	2	x	x	PUNCT
ejpam-5587	482	3	∈	∈	PROPN
ejpam-5587	482	4	ℓ2(n	ℓ2(n	PROPN
ejpam-5587	482	5	)	)	PUNCT
ejpam-5587	482	6	:	:	PUNCT
ejpam-5587	482	7	(	(	PUNCT
ejpam-5587	482	8	0	0	NUM
ejpam-5587	482	9	,	,	PUNCT
ejpam-5587	482	10	2x2	2x2	NUM
ejpam-5587	482	11	,	,	PUNCT
ejpam-5587	482	12	1	1	NUM
ejpam-5587	482	13	3x3	3x3	NUM
ejpam-5587	482	14	,	,	PUNCT
ejpam-5587	482	15	4x4	4x4	NUM
ejpam-5587	482	16	,	,	PUNCT
ejpam-5587	482	17	1	1	NUM
ejpam-5587	482	18	5x5	5x5	NUM
ejpam-5587	482	19	,	,	PUNCT
ejpam-5587	482	20	·	·	PUNCT
ejpam-5587	482	21	·	·	PUNCT
ejpam-5587	482	22	·	·	PUNCT
ejpam-5587	482	23	)	)	PUNCT
ejpam-5587	483	1	∈	∈	PROPN
ejpam-5587	483	2	ℓ2(n	ℓ2(n	PROPN
ejpam-5587	483	3	)	)	PUNCT
ejpam-5587	483	4	}	}	PUNCT
ejpam-5587	483	5	.	.	PUNCT
ejpam-5587	484	1	here	here	ADV
ejpam-5587	484	2	t	t	PROPN
ejpam-5587	484	3	is	be	AUX
ejpam-5587	484	4	a	a	DET
ejpam-5587	484	5	densely	densely	ADV
ejpam-5587	484	6	defined	define	VERB
ejpam-5587	484	7	closed	close	VERB
ejpam-5587	484	8	totally	totally	ADV
ejpam-5587	484	9	paranormal	paranormal	ADJ
ejpam-5587	484	10	operator	operator	NOUN
ejpam-5587	484	11	.	.	PUNCT
ejpam-5587	485	1	since	since	SCONJ
ejpam-5587	485	2	t	t	PROPN
ejpam-5587	485	3	is	be	AUX
ejpam-5587	485	4	not	not	PART
ejpam-5587	485	5	one	one	NUM
ejpam-5587	485	6	to	to	ADP
ejpam-5587	485	7	one	one	NUM
ejpam-5587	485	8	,	,	PUNCT
ejpam-5587	485	9	0	0	NUM
ejpam-5587	485	10	∈	∈	NOUN
ejpam-5587	485	11	σp(t	σp(t	PUNCT
ejpam-5587	485	12	)	)	PUNCT
ejpam-5587	485	13	but	but	CCONJ
ejpam-5587	485	14	it	it	PRON
ejpam-5587	485	15	is	be	AUX
ejpam-5587	485	16	not	not	PART
ejpam-5587	485	17	an	an	DET
ejpam-5587	485	18	isolated	isolated	ADJ
ejpam-5587	485	19	point	point	NOUN
ejpam-5587	485	20	of	of	ADP
ejpam-5587	485	21	the	the	DET
ejpam-5587	485	22	spectrum	spectrum	NOUN
ejpam-5587	485	23	σ(t	σ(t	PROPN
ejpam-5587	485	24	)	)	PUNCT
ejpam-5587	486	1	=	=	PRON
ejpam-5587	486	2	{	{	PUNCT
ejpam-5587	486	3	0	0	NUM
ejpam-5587	486	4	,	,	PUNCT
ejpam-5587	486	5	2	2	NUM
ejpam-5587	486	6	,	,	PUNCT
ejpam-5587	486	7	13	13	NUM
ejpam-5587	486	8	,	,	PUNCT
ejpam-5587	486	9	4	4	NUM
ejpam-5587	486	10	,	,	PUNCT
ejpam-5587	486	11	1	1	NUM
ejpam-5587	486	12	5	5	NUM
ejpam-5587	486	13	,	,	PUNCT
ejpam-5587	486	14	·	·	PUNCT
ejpam-5587	486	15	·	·	PUNCT
ejpam-5587	486	16	·	·	PUNCT
ejpam-5587	486	17	}	}	PUNCT
ejpam-5587	486	18	.	.	PUNCT
ejpam-5587	487	1	theorem	theorem	ADJ
ejpam-5587	487	2	11	11	NUM
ejpam-5587	487	3	.	.	PUNCT
ejpam-5587	488	1	suppose	suppose	VERB
ejpam-5587	488	2	t	t	PROPN
ejpam-5587	488	3	∈	∈	PROPN
ejpam-5587	488	4	l(h	l(h	PROPN
ejpam-5587	488	5	)	)	PUNCT
ejpam-5587	488	6	is	be	AUX
ejpam-5587	488	7	a	a	DET
ejpam-5587	488	8	densely	densely	ADV
ejpam-5587	488	9	defined	define	VERB
ejpam-5587	488	10	closed	close	VERB
ejpam-5587	488	11	totally	totally	ADV
ejpam-5587	488	12	paranormal	paranormal	ADJ
ejpam-5587	488	13	operator	operator	NOUN
ejpam-5587	488	14	,	,	PUNCT
ejpam-5587	488	15	ker(t	ker(t	NOUN
ejpam-5587	488	16	)	)	PUNCT
ejpam-5587	489	1	=	=	SYM
ejpam-5587	489	2	ker(t	ker(t	NOUN
ejpam-5587	489	3	∗	∗	NOUN
ejpam-5587	489	4	)	)	PUNCT
ejpam-5587	489	5	.	.	PUNCT
ejpam-5587	490	1	then	then	ADV
ejpam-5587	490	2	ran(t	ran(t	PROPN
ejpam-5587	490	3	)	)	PUNCT
ejpam-5587	490	4	is	be	AUX
ejpam-5587	490	5	closed	close	VERB
ejpam-5587	490	6	if	if	SCONJ
ejpam-5587	490	7	and	and	CCONJ
ejpam-5587	490	8	only	only	ADV
ejpam-5587	490	9	if	if	SCONJ
ejpam-5587	490	10	0	0	NUM
ejpam-5587	490	11	is	be	AUX
ejpam-5587	490	12	not	not	PART
ejpam-5587	490	13	an	an	DET
ejpam-5587	490	14	accumulation	accumulation	NOUN
ejpam-5587	490	15	point	point	NOUN
ejpam-5587	490	16	of	of	ADP
ejpam-5587	490	17	σ(t	σ(t	PROPN
ejpam-5587	490	18	)	)	PUNCT
ejpam-5587	490	19	.	.	PUNCT
ejpam-5587	491	1	proof	proof	NOUN
ejpam-5587	491	2	.	.	PUNCT
ejpam-5587	492	1	by	by	ADP
ejpam-5587	492	2	theorem	theorem	NOUN
ejpam-5587	492	3	2	2	NUM
ejpam-5587	492	4	,	,	PUNCT
ejpam-5587	492	5	ran(t	ran(t	PROPN
ejpam-5587	492	6	)	)	PUNCT
ejpam-5587	492	7	is	be	AUX
ejpam-5587	492	8	closed	close	VERB
ejpam-5587	492	9	if	if	SCONJ
ejpam-5587	492	10	and	and	CCONJ
ejpam-5587	492	11	only	only	ADV
ejpam-5587	492	12	if	if	SCONJ
ejpam-5587	492	13	γ(t	γ(t	NOUN
ejpam-5587	492	14	)	)	PUNCT
ejpam-5587	492	15	>	>	X
ejpam-5587	492	16	0	0	PUNCT
ejpam-5587	492	17	and	and	CCONJ
ejpam-5587	492	18	by	by	ADP
ejpam-5587	492	19	corollary	corollary	ADJ
ejpam-5587	492	20	2	2	NUM
ejpam-5587	492	21	,	,	PUNCT
ejpam-5587	492	22	γ(t	γ(t	NOUN
ejpam-5587	492	23	)	)	PUNCT
ejpam-5587	493	1	=	=	SYM
ejpam-5587	493	2	d(t	d(t	PROPN
ejpam-5587	493	3	)	)	PUNCT
ejpam-5587	493	4	.	.	PUNCT
ejpam-5587	494	1	hence	hence	ADV
ejpam-5587	494	2	,	,	PUNCT
ejpam-5587	494	3	ran(t	ran(t	PROPN
ejpam-5587	494	4	)	)	PUNCT
ejpam-5587	494	5	is	be	AUX
ejpam-5587	494	6	closed	close	VERB
ejpam-5587	494	7	if	if	SCONJ
ejpam-5587	494	8	and	and	CCONJ
ejpam-5587	494	9	only	only	ADV
ejpam-5587	494	10	if	if	SCONJ
ejpam-5587	494	11	d(t	d(t	PROPN
ejpam-5587	494	12	)	)	PUNCT
ejpam-5587	494	13	>	>	X
ejpam-5587	494	14	0	0	PUNCT
ejpam-5587	495	1	if	if	SCONJ
ejpam-5587	495	2	and	and	CCONJ
ejpam-5587	495	3	only	only	ADV
ejpam-5587	495	4	if	if	SCONJ
ejpam-5587	495	5	0	0	NUM
ejpam-5587	495	6	is	be	AUX
ejpam-5587	495	7	not	not	PART
ejpam-5587	495	8	an	an	DET
ejpam-5587	495	9	accumulation	accumulation	NOUN
ejpam-5587	495	10	point	point	NOUN
ejpam-5587	495	11	of	of	ADP
ejpam-5587	495	12	σ(t	σ(t	PROPN
ejpam-5587	495	13	)	)	PUNCT
ejpam-5587	495	14	.	.	PUNCT
ejpam-5587	496	1	4	4	X
ejpam-5587	496	2	.	.	X
ejpam-5587	496	3	weyl	weyl	PROPN
ejpam-5587	496	4	’s	’s	PART
ejpam-5587	496	5	theorem	theorem	NOUN
ejpam-5587	496	6	for	for	ADP
ejpam-5587	496	7	totally	totally	ADV
ejpam-5587	496	8	paranormal	paranormal	ADJ
ejpam-5587	496	9	operators	operator	NOUN
ejpam-5587	496	10	in	in	ADP
ejpam-5587	496	11	this	this	DET
ejpam-5587	496	12	section	section	NOUN
ejpam-5587	496	13	,	,	PUNCT
ejpam-5587	496	14	we	we	PRON
ejpam-5587	496	15	demonstrate	demonstrate	VERB
ejpam-5587	496	16	the	the	DET
ejpam-5587	496	17	fulfillment	fulfillment	NOUN
ejpam-5587	496	18	of	of	ADP
ejpam-5587	496	19	weyl	weyl	PROPN
ejpam-5587	496	20	’s	’s	PART
ejpam-5587	496	21	theorem	theorem	NOUN
ejpam-5587	496	22	by	by	ADP
ejpam-5587	496	23	a	a	DET
ejpam-5587	496	24	densely	densely	ADV
ejpam-5587	496	25	defined	define	VERB
ejpam-5587	496	26	closed	closed	ADJ
ejpam-5587	496	27	operator	operator	NOUN
ejpam-5587	496	28	t	t	NOUN
ejpam-5587	496	29	that	that	PRON
ejpam-5587	496	30	is	be	AUX
ejpam-5587	496	31	totally	totally	ADV
ejpam-5587	496	32	paranormal	paranormal	ADJ
ejpam-5587	496	33	.	.	PUNCT
ejpam-5587	497	1	additionally	additionally	ADV
ejpam-5587	497	2	,	,	PUNCT
ejpam-5587	497	3	we	we	PRON
ejpam-5587	497	4	establish	establish	VERB
ejpam-5587	497	5	the	the	DET
ejpam-5587	497	6	self	self	NOUN
ejpam-5587	497	7	-	-	PUNCT
ejpam-5587	497	8	adjointness	adjointness	NOUN
ejpam-5587	497	9	of	of	ADP
ejpam-5587	497	10	the	the	DET
ejpam-5587	497	11	riesz	riesz	PROPN
ejpam-5587	497	12	projection	projection	NOUN
ejpam-5587	497	13	eµ	eµ	NOUN
ejpam-5587	497	14	corresponding	correspond	VERB
ejpam-5587	497	15	to	to	ADP
ejpam-5587	497	16	any	any	DET
ejpam-5587	497	17	non	non	ADJ
ejpam-5587	497	18	-	-	ADJ
ejpam-5587	497	19	zero	zero	NUM
ejpam-5587	497	20	isolated	isolate	VERB
ejpam-5587	497	21	spectral	spectral	ADJ
ejpam-5587	497	22	value	value	NOUN
ejpam-5587	497	23	µ	µ	PROPN
ejpam-5587	497	24	of	of	ADP
ejpam-5587	497	25	t	t	PROPN
ejpam-5587	497	26	.	.	PUNCT
ejpam-5587	498	1	for	for	SCONJ
ejpam-5587	498	2	a	a	DET
ejpam-5587	498	3	hilbert	hilbert	NOUN
ejpam-5587	498	4	space	space	NOUN
ejpam-5587	498	5	h	h	NOUN
ejpam-5587	498	6	decomposed	decompose	VERB
ejpam-5587	498	7	as	as	ADP
ejpam-5587	498	8	h	h	NOUN
ejpam-5587	498	9	=	=	SYM
ejpam-5587	498	10	h1⊕h2	h1⊕h2	PROPN
ejpam-5587	498	11	,	,	PUNCT
ejpam-5587	498	12	where	where	SCONJ
ejpam-5587	498	13	t	t	PROPN
ejpam-5587	498	14	∈	∈	PROPN
ejpam-5587	498	15	l(h	l(h	PROPN
ejpam-5587	498	16	)	)	PUNCT
ejpam-5587	498	17	is	be	AUX
ejpam-5587	498	18	a	a	DET
ejpam-5587	498	19	closed	closed	ADJ
ejpam-5587	498	20	operator	operator	NOUN
ejpam-5587	498	21	,	,	PUNCT
ejpam-5587	498	22	we	we	PRON
ejpam-5587	498	23	ascertain	ascertain	VERB
ejpam-5587	498	24	the	the	DET
ejpam-5587	498	25	block	block	NOUN
ejpam-5587	498	26	matrix	matrix	NOUN
ejpam-5587	498	27	representation	representation	NOUN
ejpam-5587	498	28	of	of	ADP
ejpam-5587	498	29	t	t	PROPN
ejpam-5587	498	30	.	.	PUNCT
ejpam-5587	499	1	t	t	PROPN
ejpam-5587	500	1	=	=	PUNCT
ejpam-5587	501	1	[	[	PUNCT
ejpam-5587	501	2	t11	t11	PROPN
ejpam-5587	501	3	t12	t12	PROPN
ejpam-5587	501	4	t21	t21	PROPN
ejpam-5587	501	5	t22	t22	PROPN
ejpam-5587	501	6	]	]	PUNCT
ejpam-5587	501	7	(	(	PUNCT
ejpam-5587	501	8	4	4	X
ejpam-5587	501	9	)	)	PUNCT
ejpam-5587	501	10	s.	s.	PROPN
ejpam-5587	501	11	alnabulsi	alnabulsi	PROPN
ejpam-5587	501	12	,	,	PUNCT
ejpam-5587	501	13	m.h.m	m.h.m	PROPN
ejpam-5587	501	14	.	.	PUNCT
ejpam-5587	501	15	rashid	rashid	PROPN
ejpam-5587	501	16	/	/	SYM
ejpam-5587	501	17	eur	eur	PROPN
ejpam-5587	501	18	.	.	PUNCT
ejpam-5587	502	1	j.	j.	PROPN
ejpam-5587	502	2	pure	pure	PROPN
ejpam-5587	502	3	appl	appl	PROPN
ejpam-5587	502	4	.	.	PROPN
ejpam-5587	502	5	math	math	PROPN
ejpam-5587	502	6	,	,	PUNCT
ejpam-5587	502	7	18	18	NUM
ejpam-5587	502	8	(	(	PUNCT
ejpam-5587	502	9	1	1	NUM
ejpam-5587	502	10	)	)	PUNCT
ejpam-5587	502	11	(	(	PUNCT
ejpam-5587	502	12	2025	2025	NUM
ejpam-5587	502	13	)	)	PUNCT
ejpam-5587	502	14	,	,	PUNCT
ejpam-5587	502	15	5587	5587	NUM
ejpam-5587	502	16	14	14	NUM
ejpam-5587	502	17	of	of	ADP
ejpam-5587	502	18	20	20	NUM
ejpam-5587	502	19	where	where	SCONJ
ejpam-5587	502	20	tij	tij	NOUN
ejpam-5587	502	21	:	:	PUNCT
ejpam-5587	502	22	d(t	d(t	PROPN
ejpam-5587	502	23	)	)	PUNCT
ejpam-5587	502	24	∩	∩	PROPN
ejpam-5587	502	25	hj	hj	PROPN
ejpam-5587	502	26	→	→	AUX
ejpam-5587	502	27	hi	hi	PROPN
ejpam-5587	502	28	is	be	AUX
ejpam-5587	502	29	defined	define	VERB
ejpam-5587	502	30	by	by	ADP
ejpam-5587	502	31	tij	tij	PROPN
ejpam-5587	502	32	=	=	SYM
ejpam-5587	502	33	phitphj	phitphj	PROPN
ejpam-5587	502	34	|d(t	|d(t	ADJ
ejpam-5587	502	35	)	)	PUNCT
ejpam-5587	502	36	∩hj	∩hj	NOUN
ejpam-5587	502	37	for	for	ADP
ejpam-5587	502	38	i	i	PROPN
ejpam-5587	502	39	,	,	PUNCT
ejpam-5587	502	40	j	j	PROPN
ejpam-5587	502	41	=	=	SYM
ejpam-5587	502	42	1	1	NUM
ejpam-5587	502	43	,	,	PUNCT
ejpam-5587	502	44	2	2	NUM
ejpam-5587	502	45	.	.	X
ejpam-5587	503	1	here	here	ADV
ejpam-5587	503	2	phi	phi	NOUN
ejpam-5587	503	3	is	be	AUX
ejpam-5587	503	4	an	an	DET
ejpam-5587	503	5	orthogonal	orthogonal	ADJ
ejpam-5587	503	6	projection	projection	NOUN
ejpam-5587	503	7	onto	onto	ADP
ejpam-5587	503	8	hi	hi	INTJ
ejpam-5587	503	9	.	.	PUNCT
ejpam-5587	504	1	for	for	ADP
ejpam-5587	504	2	(	(	PUNCT
ejpam-5587	504	3	x1	x1	PROPN
ejpam-5587	504	4	,	,	PUNCT
ejpam-5587	504	5	x2	x2	ADJ
ejpam-5587	504	6	)	)	PUNCT
ejpam-5587	504	7	∈	∈	PROPN
ejpam-5587	504	8	(	(	PUNCT
ejpam-5587	504	9	h1	h1	PROPN
ejpam-5587	504	10	∩d(t	∩d(t	ADJ
ejpam-5587	504	11	)	)	PUNCT
ejpam-5587	504	12	⊕	⊕	PROPN
ejpam-5587	504	13	(	(	PUNCT
ejpam-5587	504	14	h2	h2	PROPN
ejpam-5587	504	15	∩d(t	∩d(t	PROPN
ejpam-5587	504	16	)	)	PUNCT
ejpam-5587	504	17	)	)	PUNCT
ejpam-5587	504	18	,	,	PUNCT
ejpam-5587	504	19	t	t	PROPN
ejpam-5587	504	20	(	(	PUNCT
ejpam-5587	504	21	x1	x1	PROPN
ejpam-5587	504	22	,	,	PUNCT
ejpam-5587	504	23	x2	x2	PROPN
ejpam-5587	504	24	)	)	PUNCT
ejpam-5587	504	25	=	=	SYM
ejpam-5587	504	26	(	(	PUNCT
ejpam-5587	504	27	t11x1	t11x1	NOUN
ejpam-5587	504	28	+	+	CCONJ
ejpam-5587	504	29	t12x2	t12x2	PROPN
ejpam-5587	504	30	,	,	PUNCT
ejpam-5587	504	31	t21x1	t21x1	PROPN
ejpam-5587	504	32	+	+	CCONJ
ejpam-5587	504	33	t22x2	t22x2	NOUN
ejpam-5587	504	34	)	)	PUNCT
ejpam-5587	504	35	.	.	PUNCT
ejpam-5587	505	1	remark	remark	PROPN
ejpam-5587	505	2	3	3	NUM
ejpam-5587	505	3	.	.	PUNCT
ejpam-5587	506	1	let	let	VERB
ejpam-5587	506	2	t	t	PROPN
ejpam-5587	506	3	be	be	AUX
ejpam-5587	506	4	as	as	ADV
ejpam-5587	506	5	defined	define	VERB
ejpam-5587	506	6	in	in	ADP
ejpam-5587	506	7	equation	equation	NOUN
ejpam-5587	506	8	(	(	PUNCT
ejpam-5587	506	9	4	4	NUM
ejpam-5587	506	10	)	)	PUNCT
ejpam-5587	506	11	.	.	PUNCT
ejpam-5587	507	1	if	if	SCONJ
ejpam-5587	507	2	h1	h1	NOUN
ejpam-5587	507	3	=	=	PUNCT
ejpam-5587	507	4	ker(t	ker(t	NOUN
ejpam-5587	507	5	)	)	PUNCT
ejpam-5587	507	6	̸=	̸=	PROPN
ejpam-5587	507	7	{	{	PUNCT
ejpam-5587	507	8	0	0	NUM
ejpam-5587	507	9	}	}	PUNCT
ejpam-5587	507	10	and	and	CCONJ
ejpam-5587	507	11	h2	h2	NOUN
ejpam-5587	507	12	=	=	SYM
ejpam-5587	507	13	ker(t	ker(t	PROPN
ejpam-5587	507	14	)	)	PUNCT
ejpam-5587	507	15	⊥	⊥	NOUN
ejpam-5587	507	16	,	,	PUNCT
ejpam-5587	507	17	then	then	ADV
ejpam-5587	507	18	t	t	PROPN
ejpam-5587	507	19	=	=	PUNCT
ejpam-5587	507	20	[	[	PUNCT
ejpam-5587	507	21	0	0	NUM
ejpam-5587	507	22	t12	t12	PROPN
ejpam-5587	507	23	0	0	NUM
ejpam-5587	507	24	t22	t22	PROPN
ejpam-5587	507	25	]	]	PUNCT
ejpam-5587	507	26	.	.	PUNCT
ejpam-5587	508	1	(	(	PUNCT
ejpam-5587	508	2	5	5	NUM
ejpam-5587	508	3	)	)	PUNCT
ejpam-5587	508	4	•	•	NOUN
ejpam-5587	508	5	if	if	SCONJ
ejpam-5587	508	6	t	t	PROPN
ejpam-5587	508	7	is	be	AUX
ejpam-5587	508	8	densely	densely	ADV
ejpam-5587	508	9	defined	define	VERB
ejpam-5587	508	10	closed	closed	ADJ
ejpam-5587	508	11	operator	operator	NOUN
ejpam-5587	508	12	then	then	ADV
ejpam-5587	508	13	by	by	ADP
ejpam-5587	508	14	lemma	lemma	PROPN
ejpam-5587	508	15	1	1	NUM
ejpam-5587	508	16	,	,	PUNCT
ejpam-5587	508	17	t22	t22	PROPN
ejpam-5587	508	18	∈	∈	PROPN
ejpam-5587	508	19	l(ker(t	l(ker(t	NOUN
ejpam-5587	508	20	)	)	PUNCT
ejpam-5587	508	21	⊥	⊥	X
ejpam-5587	508	22	)	)	PUNCT
ejpam-5587	508	23	is	be	AUX
ejpam-5587	508	24	also	also	ADV
ejpam-5587	508	25	densely	densely	ADV
ejpam-5587	508	26	defined	define	VERB
ejpam-5587	508	27	closed	closed	ADJ
ejpam-5587	508	28	operator	operator	NOUN
ejpam-5587	508	29	.	.	PUNCT
ejpam-5587	509	1	•	•	ADP
ejpam-5587	509	2	it	it	PRON
ejpam-5587	509	3	can	can	AUX
ejpam-5587	509	4	be	be	AUX
ejpam-5587	509	5	easily	easily	ADV
ejpam-5587	509	6	checked	check	VERB
ejpam-5587	509	7	that	that	PRON
ejpam-5587	509	8	ran(t22	ran(t22	PROPN
ejpam-5587	509	9	)	)	PUNCT
ejpam-5587	509	10	=	=	SYM
ejpam-5587	509	11	ran(t	ran(t	NOUN
ejpam-5587	509	12	)	)	PUNCT
ejpam-5587	509	13	∩	∩	ADJ
ejpam-5587	509	14	ker(t	ker(t	NOUN
ejpam-5587	509	15	)	)	PUNCT
ejpam-5587	509	16	⊥.	⊥.	PROPN
ejpam-5587	509	17	if	if	SCONJ
ejpam-5587	509	18	ran(t	ran(t	NOUN
ejpam-5587	509	19	)	)	PUNCT
ejpam-5587	509	20	is	be	AUX
ejpam-5587	509	21	closed	close	VERB
ejpam-5587	509	22	,	,	PUNCT
ejpam-5587	509	23	then	then	ADV
ejpam-5587	509	24	ran(t22	ran(t22	X
ejpam-5587	509	25	)	)	PUNCT
ejpam-5587	509	26	is	be	AUX
ejpam-5587	509	27	closed	close	VERB
ejpam-5587	509	28	in	in	ADP
ejpam-5587	509	29	ker(t	ker(t	NOUN
ejpam-5587	509	30	)	)	PUNCT
ejpam-5587	509	31	⊥.	⊥.	NUM
ejpam-5587	510	1	we	we	PRON
ejpam-5587	510	2	say	say	VERB
ejpam-5587	510	3	a	a	DET
ejpam-5587	510	4	closed	closed	ADJ
ejpam-5587	510	5	operator	operator	NOUN
ejpam-5587	510	6	t	t	PROPN
ejpam-5587	510	7	∈	∈	PROPN
ejpam-5587	510	8	l(h	l(h	PROPN
ejpam-5587	510	9	)	)	PUNCT
ejpam-5587	510	10	satisfy	satisfy	VERB
ejpam-5587	510	11	the	the	DET
ejpam-5587	510	12	weyl	weyl	PROPN
ejpam-5587	510	13	’s	’s	PART
ejpam-5587	510	14	theorem	theorem	NOUN
ejpam-5587	510	15	if	if	SCONJ
ejpam-5587	510	16	the	the	DET
ejpam-5587	510	17	weyl	weyl	PROPN
ejpam-5587	510	18	’s	’s	PART
ejpam-5587	510	19	spectrum	spectrum	NOUN
ejpam-5587	510	20	,	,	PUNCT
ejpam-5587	510	21	σw(t	σw(t	PUNCT
ejpam-5587	510	22	)	)	PUNCT
ejpam-5587	510	23	consists	consist	VERB
ejpam-5587	510	24	of	of	ADP
ejpam-5587	510	25	all	all	DET
ejpam-5587	510	26	spectral	spectral	ADJ
ejpam-5587	510	27	values	value	NOUN
ejpam-5587	510	28	of	of	ADP
ejpam-5587	510	29	t	t	PROPN
ejpam-5587	510	30	except	except	SCONJ
ejpam-5587	510	31	the	the	DET
ejpam-5587	510	32	isolated	isolate	VERB
ejpam-5587	510	33	eigenvalues	eigenvalue	NOUN
ejpam-5587	510	34	of	of	ADP
ejpam-5587	510	35	finite	finite	ADJ
ejpam-5587	510	36	multiplicity	multiplicity	NOUN
ejpam-5587	510	37	.	.	PUNCT
ejpam-5587	511	1	that	that	PRON
ejpam-5587	511	2	is	be	AUX
ejpam-5587	511	3	,	,	PUNCT
ejpam-5587	511	4	σ(t	σ(t	PROPN
ejpam-5587	511	5	)	)	PUNCT
ejpam-5587	511	6	\	\	PROPN
ejpam-5587	511	7	σw(t	σw(t	PUNCT
ejpam-5587	511	8	)	)	PUNCT
ejpam-5587	512	1	=	=	SYM
ejpam-5587	512	2	π00(t	π00(t	ADJ
ejpam-5587	512	3	)	)	PUNCT
ejpam-5587	512	4	.	.	PUNCT
ejpam-5587	513	1	coburn	coburn	NOUN
ejpam-5587	514	1	[	[	X
ejpam-5587	514	2	5	5	X
ejpam-5587	514	3	]	]	PUNCT
ejpam-5587	514	4	established	establish	VERB
ejpam-5587	514	5	that	that	SCONJ
ejpam-5587	514	6	weyl	weyl	PROPN
ejpam-5587	514	7	’s	’s	PART
ejpam-5587	514	8	theorem	theorem	PROPN
ejpam-5587	514	9	applies	applie	NOUN
ejpam-5587	514	10	to	to	ADP
ejpam-5587	514	11	all	all	DET
ejpam-5587	514	12	bounded	bound	VERB
ejpam-5587	514	13	hyponormal	hyponormal	ADJ
ejpam-5587	514	14	and	and	CCONJ
ejpam-5587	514	15	toeplitz	toeplitz	NOUN
ejpam-5587	514	16	operators	operator	NOUN
ejpam-5587	514	17	.	.	PUNCT
ejpam-5587	515	1	schmoeger	schmoeger	NOUN
ejpam-5587	516	1	[	[	X
ejpam-5587	516	2	22	22	NUM
ejpam-5587	516	3	]	]	PUNCT
ejpam-5587	516	4	subsequently	subsequently	ADV
ejpam-5587	516	5	extended	extend	VERB
ejpam-5587	516	6	this	this	DET
ejpam-5587	516	7	finding	finding	NOUN
ejpam-5587	516	8	to	to	PART
ejpam-5587	516	9	include	include	VERB
ejpam-5587	516	10	bounded	bound	VERB
ejpam-5587	516	11	totally	totally	ADV
ejpam-5587	516	12	paranormal	paranormal	ADJ
ejpam-5587	516	13	operators	operator	NOUN
ejpam-5587	516	14	.	.	PUNCT
ejpam-5587	517	1	in	in	ADP
ejpam-5587	517	2	this	this	DET
ejpam-5587	517	3	study	study	NOUN
ejpam-5587	517	4	,	,	PUNCT
ejpam-5587	517	5	we	we	PRON
ejpam-5587	517	6	seek	seek	VERB
ejpam-5587	517	7	to	to	PART
ejpam-5587	517	8	verify	verify	VERB
ejpam-5587	517	9	the	the	DET
ejpam-5587	517	10	applicability	applicability	NOUN
ejpam-5587	517	11	of	of	ADP
ejpam-5587	517	12	weyl	weyl	PROPN
ejpam-5587	517	13	’s	’s	PART
ejpam-5587	517	14	theorem	theorem	NOUN
ejpam-5587	517	15	to	to	PART
ejpam-5587	517	16	unbounded	unbounded	ADJ
ejpam-5587	517	17	totally	totally	ADV
ejpam-5587	517	18	paranormal	paranormal	ADJ
ejpam-5587	517	19	operators	operator	NOUN
ejpam-5587	517	20	.	.	PUNCT
ejpam-5587	518	1	theorem	theorem	NOUN
ejpam-5587	518	2	12	12	NUM
ejpam-5587	518	3	.	.	PUNCT
ejpam-5587	519	1	let	let	VERB
ejpam-5587	519	2	t	t	PROPN
ejpam-5587	519	3	∈	∈	PROPN
ejpam-5587	519	4	l(h	l(h	PROPN
ejpam-5587	519	5	)	)	PUNCT
ejpam-5587	519	6	be	be	AUX
ejpam-5587	519	7	a	a	DET
ejpam-5587	519	8	densely	densely	ADV
ejpam-5587	519	9	defined	define	VERB
ejpam-5587	519	10	closed	close	VERB
ejpam-5587	519	11	totally	totally	ADV
ejpam-5587	519	12	paranormal	paranormal	ADJ
ejpam-5587	519	13	operator	operator	NOUN
ejpam-5587	519	14	.	.	PUNCT
ejpam-5587	520	1	then	then	ADV
ejpam-5587	520	2	weyl	weyl	PROPN
ejpam-5587	520	3	’s	’s	PART
ejpam-5587	520	4	theorem	theorem	NOUN
ejpam-5587	520	5	holds	hold	VERB
ejpam-5587	520	6	for	for	ADP
ejpam-5587	520	7	t	t	NOUN
ejpam-5587	520	8	,	,	PUNCT
ejpam-5587	520	9	that	that	ADV
ejpam-5587	520	10	is	is	ADV
ejpam-5587	520	11	,	,	PUNCT
ejpam-5587	520	12	σ(t	σ(t	PROPN
ejpam-5587	520	13	)	)	PUNCT
ejpam-5587	520	14	\	\	PROPN
ejpam-5587	520	15	σw(t	σw(t	PUNCT
ejpam-5587	520	16	)	)	PUNCT
ejpam-5587	521	1	=	=	SYM
ejpam-5587	521	2	π00(t	π00(t	ADJ
ejpam-5587	521	3	)	)	PUNCT
ejpam-5587	521	4	.	.	PUNCT
ejpam-5587	522	1	proof	proof	NOUN
ejpam-5587	522	2	.	.	PUNCT
ejpam-5587	523	1	let	let	VERB
ejpam-5587	523	2	µ	µ	X
ejpam-5587	523	3	∈	∈	PRON
ejpam-5587	523	4	σ(t	σ(t	PROPN
ejpam-5587	523	5	)	)	PUNCT
ejpam-5587	524	1	\σw(t	\σw(t	ADV
ejpam-5587	524	2	)	)	PUNCT
ejpam-5587	524	3	.	.	PUNCT
ejpam-5587	525	1	so	so	ADV
ejpam-5587	525	2	,	,	PUNCT
ejpam-5587	525	3	we	we	PRON
ejpam-5587	525	4	have	have	VERB
ejpam-5587	525	5	dim(ker(t−µi	dim(ker(t−µi	NOUN
ejpam-5587	525	6	)	)	PUNCT
ejpam-5587	525	7	)	)	PUNCT
ejpam-5587	526	1	=	=	PUNCT
ejpam-5587	526	2	dim(ker(t−µi)∗	dim(ker(t−µi)∗	X
ejpam-5587	526	3	)	)	PUNCT
ejpam-5587	527	1	<	<	X
ejpam-5587	528	1	+	+	X
ejpam-5587	528	2	∞	∞	NUM
ejpam-5587	528	3	and	and	CCONJ
ejpam-5587	528	4	ran(t	ran(t	PROPN
ejpam-5587	528	5	−	−	PROPN
ejpam-5587	528	6	µi	µi	PROPN
ejpam-5587	528	7	)	)	PUNCT
ejpam-5587	528	8	is	be	AUX
ejpam-5587	528	9	closed	close	VERB
ejpam-5587	528	10	.	.	PUNCT
ejpam-5587	529	1	on	on	ADP
ejpam-5587	529	2	h	h	NOUN
ejpam-5587	529	3	=	=	NOUN
ejpam-5587	529	4	ker(t	ker(t	NOUN
ejpam-5587	529	5	−	−	NOUN
ejpam-5587	529	6	µi)⊕	µi)⊕	SYM
ejpam-5587	529	7	ker(t	ker(t	NOUN
ejpam-5587	529	8	−	−	PROPN
ejpam-5587	529	9	µi)⊥	µi)⊥	PROPN
ejpam-5587	529	10	,	,	PUNCT
ejpam-5587	529	11	t	t	PROPN
ejpam-5587	529	12	−	−	PROPN
ejpam-5587	530	1	µi	µi	PROPN
ejpam-5587	530	2	can	can	AUX
ejpam-5587	530	3	be	be	AUX
ejpam-5587	530	4	decomposed	decompose	VERB
ejpam-5587	530	5	as	as	ADP
ejpam-5587	530	6	t	t	NOUN
ejpam-5587	530	7	−	−	NOUN
ejpam-5587	530	8	µi	µi	INTJ
ejpam-5587	530	9	=	=	PUNCT
ejpam-5587	530	10	[	[	PUNCT
ejpam-5587	530	11	0	0	NUM
ejpam-5587	530	12	t12	t12	PROPN
ejpam-5587	530	13	0	0	NUM
ejpam-5587	530	14	t22	t22	PROPN
ejpam-5587	530	15	−	−	PROPN
ejpam-5587	530	16	µi|ker(t−µi)⊥	µi|ker(t−µi)⊥	PROPN
ejpam-5587	530	17	]	]	PUNCT
ejpam-5587	530	18	,	,	PUNCT
ejpam-5587	530	19	where	where	SCONJ
ejpam-5587	530	20	t22	t22	PROPN
ejpam-5587	530	21	=	=	PROPN
ejpam-5587	530	22	p	p	PROPN
ejpam-5587	530	23	|ker(t−µi)⊥t	|ker(t−µi)⊥t	PROPN
ejpam-5587	530	24	|ker(t−µi)⊥	|ker(t−µi)⊥	PROPN
ejpam-5587	530	25	.	.	PUNCT
ejpam-5587	531	1	by	by	ADP
ejpam-5587	531	2	remark	remark	NOUN
ejpam-5587	531	3	3	3	NUM
ejpam-5587	531	4	,	,	PUNCT
ejpam-5587	531	5	t22−µiker(t−µi)⊥	t22−µiker(t−µi)⊥	PRON
ejpam-5587	531	6	is	be	AUX
ejpam-5587	531	7	a	a	DET
ejpam-5587	531	8	densely	densely	ADV
ejpam-5587	531	9	defined	define	VERB
ejpam-5587	531	10	closed	closed	ADJ
ejpam-5587	531	11	operator	operator	NOUN
ejpam-5587	531	12	with	with	ADP
ejpam-5587	531	13	domaind(t−µi)∩ker(t−µi)⊥	domaind(t−µi)∩ker(t−µi)⊥	NOUN
ejpam-5587	531	14	and	and	CCONJ
ejpam-5587	531	15	ran(t22−µiker(t−µi)⊥	ran(t22−µiker(t−µi)⊥	NOUN
ejpam-5587	531	16	)	)	PUNCT
ejpam-5587	531	17	is	be	AUX
ejpam-5587	531	18	closed	close	VERB
ejpam-5587	531	19	.	.	PUNCT
ejpam-5587	532	1	as	as	ADP
ejpam-5587	532	2	ker(t	ker(t	NOUN
ejpam-5587	532	3	−µi	−µi	NOUN
ejpam-5587	532	4	)	)	PUNCT
ejpam-5587	532	5	is	be	AUX
ejpam-5587	532	6	finite	finite	ADJ
ejpam-5587	532	7	dimensional	dimensional	ADJ
ejpam-5587	532	8	,	,	PUNCT
ejpam-5587	532	9	this	this	PRON
ejpam-5587	532	10	implies	imply	VERB
ejpam-5587	532	11	t12	t12	PROPN
ejpam-5587	532	12	is	be	AUX
ejpam-5587	532	13	finite	finite	ADJ
ejpam-5587	532	14	rank	rank	NOUN
ejpam-5587	532	15	operator	operator	NOUN
ejpam-5587	532	16	and	and	CCONJ
ejpam-5587	532	17	by	by	ADP
ejpam-5587	532	18	remark	remark	NOUN
ejpam-5587	532	19	2	2	NUM
ejpam-5587	532	20	,	,	PUNCT
ejpam-5587	532	21	ind(t	ind(t	ADJ
ejpam-5587	532	22	−	−	PROPN
ejpam-5587	532	23	µi	µi	NOUN
ejpam-5587	532	24	)	)	PUNCT
ejpam-5587	532	25	=	=	SYM
ejpam-5587	532	26	ind	ind	NOUN
ejpam-5587	532	27	(	(	PUNCT
ejpam-5587	532	28	t22	t22	NOUN
ejpam-5587	532	29	−	−	PROPN
ejpam-5587	532	30	µi|ker(t−µi)⊥	µi|ker(t−µi)⊥	NOUN
ejpam-5587	532	31	)	)	PUNCT
ejpam-5587	532	32	=	=	SYM
ejpam-5587	533	1	0	0	X
ejpam-5587	533	2	.	.	PUNCT
ejpam-5587	534	1	since	since	SCONJ
ejpam-5587	534	2	ker(t22	ker(t22	PROPN
ejpam-5587	534	3	−	−	PROPN
ejpam-5587	534	4	µi|ker(t−µi)⊥	µi|ker(t−µi)⊥	NOUN
ejpam-5587	534	5	)	)	PUNCT
ejpam-5587	534	6	=	=	PUNCT
ejpam-5587	534	7	{	{	PUNCT
ejpam-5587	534	8	0	0	NUM
ejpam-5587	534	9	}	}	PUNCT
ejpam-5587	534	10	and	and	CCONJ
ejpam-5587	534	11	ind	ind	NOUN
ejpam-5587	534	12	(	(	PUNCT
ejpam-5587	534	13	t22	t22	NOUN
ejpam-5587	534	14	−	−	PROPN
ejpam-5587	534	15	µi|ker(t−µi)⊥	µi|ker(t−µi)⊥	NOUN
ejpam-5587	534	16	)	)	PUNCT
ejpam-5587	534	17	=	=	SYM
ejpam-5587	534	18	0	0	NUM
ejpam-5587	534	19	,	,	PUNCT
ejpam-5587	534	20	we	we	PRON
ejpam-5587	534	21	get	get	VERB
ejpam-5587	534	22	ker(t22	ker(t22	PRON
ejpam-5587	534	23	−	−	PROPN
ejpam-5587	534	24	µi|ker(t−µi)⊥	µi|ker(t−µi)⊥	NOUN
ejpam-5587	534	25	)	)	PUNCT
ejpam-5587	534	26	∗	∗	NOUN
ejpam-5587	534	27	=	=	SYM
ejpam-5587	534	28	{	{	PUNCT
ejpam-5587	534	29	0	0	NUM
ejpam-5587	534	30	}	}	PUNCT
ejpam-5587	534	31	and	and	CCONJ
ejpam-5587	534	32	consequently	consequently	ADV
ejpam-5587	534	33	ran(t22	ran(t22	VERB
ejpam-5587	534	34	−	−	PROPN
ejpam-5587	534	35	µi|⊥ker(t−µi	µi|⊥ker(t−µi	NOUN
ejpam-5587	534	36	)	)	PUNCT
ejpam-5587	534	37	)	)	PUNCT
ejpam-5587	535	1	=	=	PUNCT
ejpam-5587	535	2	ker(t	ker(t	NOUN
ejpam-5587	535	3	−	−	NOUN
ejpam-5587	535	4	µi)⊥.	µi)⊥.	PRON
ejpam-5587	535	5	thus	thus	ADV
ejpam-5587	535	6	t22	t22	PROPN
ejpam-5587	535	7	−	−	PROPN
ejpam-5587	535	8	µi|⊥ker(t−µi	µi|⊥ker(t−µi	PRON
ejpam-5587	535	9	)	)	PUNCT
ejpam-5587	535	10	has	have	AUX
ejpam-5587	535	11	bounded	bound	VERB
ejpam-5587	535	12	inverse	inverse	NOUN
ejpam-5587	535	13	and	and	CCONJ
ejpam-5587	535	14	hence	hence	ADV
ejpam-5587	535	15	µ	µ	ADV
ejpam-5587	535	16	/∈	/∈	PUNCT
ejpam-5587	535	17	σ(t22	σ(t22	NUM
ejpam-5587	535	18	)	)	PUNCT
ejpam-5587	535	19	.	.	PUNCT
ejpam-5587	536	1	as	as	ADP
ejpam-5587	536	2	σ(t	σ(t	PROPN
ejpam-5587	536	3	)	)	PUNCT
ejpam-5587	536	4	⊆	⊆	X
ejpam-5587	536	5	{	{	PUNCT
ejpam-5587	536	6	µ	µ	NOUN
ejpam-5587	536	7	}	}	PUNCT
ejpam-5587	536	8	∪	∪	ADJ
ejpam-5587	536	9	σ(t22	σ(t22	X
ejpam-5587	536	10	)	)	PUNCT
ejpam-5587	536	11	,	,	PUNCT
ejpam-5587	536	12	this	this	PRON
ejpam-5587	536	13	implies	imply	VERB
ejpam-5587	536	14	that	that	SCONJ
ejpam-5587	536	15	µ	µ	NOUN
ejpam-5587	536	16	is	be	AUX
ejpam-5587	536	17	an	an	DET
ejpam-5587	536	18	isolated	isolated	ADJ
ejpam-5587	536	19	point	point	NOUN
ejpam-5587	536	20	of	of	ADP
ejpam-5587	536	21	σ(t	σ(t	PROPN
ejpam-5587	536	22	)	)	PUNCT
ejpam-5587	536	23	.	.	PUNCT
ejpam-5587	537	1	hence	hence	ADV
ejpam-5587	537	2	µ	µ	X
ejpam-5587	537	3	∈	∈	NOUN
ejpam-5587	537	4	π00(t	π00(t	X
ejpam-5587	537	5	)	)	PUNCT
ejpam-5587	537	6	.	.	PUNCT
ejpam-5587	538	1	conversely	conversely	ADV
ejpam-5587	538	2	,	,	PUNCT
ejpam-5587	538	3	let	let	VERB
ejpam-5587	538	4	µ	µ	PRON
ejpam-5587	538	5	∈	∈	NOUN
ejpam-5587	538	6	π00(t	π00(t	X
ejpam-5587	538	7	)	)	PUNCT
ejpam-5587	538	8	.	.	PUNCT
ejpam-5587	539	1	now	now	ADV
ejpam-5587	539	2	consider	consider	VERB
ejpam-5587	539	3	the	the	DET
ejpam-5587	539	4	riesz	riesz	NOUN
ejpam-5587	539	5	projection	projection	NOUN
ejpam-5587	539	6	eµ	eµ	NOUN
ejpam-5587	539	7	with	with	ADP
ejpam-5587	539	8	respect	respect	NOUN
ejpam-5587	539	9	to	to	ADP
ejpam-5587	539	10	µ.	µ.	NOUN
ejpam-5587	539	11	by	by	ADP
ejpam-5587	539	12	theorem	theorem	NOUN
ejpam-5587	539	13	1	1	NUM
ejpam-5587	539	14	and	and	CCONJ
ejpam-5587	539	15	s.	s.	PROPN
ejpam-5587	539	16	alnabulsi	alnabulsi	PROPN
ejpam-5587	539	17	,	,	PUNCT
ejpam-5587	539	18	m.h.m	m.h.m	PROPN
ejpam-5587	539	19	.	.	PUNCT
ejpam-5587	539	20	rashid	rashid	PROPN
ejpam-5587	539	21	/	/	SYM
ejpam-5587	539	22	eur	eur	PROPN
ejpam-5587	539	23	.	.	PUNCT
ejpam-5587	540	1	j.	j.	PROPN
ejpam-5587	540	2	pure	pure	PROPN
ejpam-5587	540	3	appl	appl	PROPN
ejpam-5587	540	4	.	.	PROPN
ejpam-5587	540	5	math	math	PROPN
ejpam-5587	540	6	,	,	PUNCT
ejpam-5587	540	7	18	18	NUM
ejpam-5587	540	8	(	(	PUNCT
ejpam-5587	540	9	1	1	NUM
ejpam-5587	540	10	)	)	PUNCT
ejpam-5587	540	11	(	(	PUNCT
ejpam-5587	540	12	2025	2025	NUM
ejpam-5587	540	13	)	)	PUNCT
ejpam-5587	540	14	,	,	PUNCT
ejpam-5587	540	15	5587	5587	NUM
ejpam-5587	540	16	15	15	NUM
ejpam-5587	540	17	of	of	ADP
ejpam-5587	540	18	20	20	NUM
ejpam-5587	540	19	theorem	theorem	VERB
ejpam-5587	540	20	7	7	NUM
ejpam-5587	540	21	,	,	PUNCT
ejpam-5587	540	22	µ	µ	X
ejpam-5587	540	23	/∈	/∈	PUNCT
ejpam-5587	541	1	σ(t	σ(t	PROPN
ejpam-5587	541	2	|ker(eµ	|ker(eµ	PROPN
ejpam-5587	541	3	)	)	PUNCT
ejpam-5587	541	4	)	)	PUNCT
ejpam-5587	542	1	and	and	CCONJ
ejpam-5587	542	2	ran(t	ran(t	NUM
ejpam-5587	542	3	−	−	PROPN
ejpam-5587	542	4	µi	µi	PROPN
ejpam-5587	542	5	)	)	PUNCT
ejpam-5587	542	6	=	=	PRON
ejpam-5587	542	7	ran	run	VERB
ejpam-5587	542	8	(	(	PUNCT
ejpam-5587	542	9	(	(	PUNCT
ejpam-5587	542	10	t	t	PROPN
ejpam-5587	542	11	−	−	PROPN
ejpam-5587	542	12	µi	µi	PROPN
ejpam-5587	542	13	)	)	PUNCT
ejpam-5587	542	14	|ker(eµ	|ker(eµ	PROPN
ejpam-5587	542	15	)	)	PUNCT
ejpam-5587	542	16	)	)	PUNCT
ejpam-5587	543	1	=	=	SYM
ejpam-5587	543	2	ker(eµ	ker(eµ	NOUN
ejpam-5587	543	3	)	)	PUNCT
ejpam-5587	543	4	.	.	PUNCT
ejpam-5587	544	1	since	since	SCONJ
ejpam-5587	544	2	µ	µ	PRON
ejpam-5587	544	3	/∈	/∈	PUNCT
ejpam-5587	544	4	σ(t	σ(t	PROPN
ejpam-5587	544	5	|ker(eµ	|ker(eµ	PROPN
ejpam-5587	544	6	)	)	PUNCT
ejpam-5587	544	7	)	)	PUNCT
ejpam-5587	544	8	,	,	PUNCT
ejpam-5587	544	9	we	we	PRON
ejpam-5587	544	10	have	have	VERB
ejpam-5587	544	11	that	that	DET
ejpam-5587	544	12	ran((t	ran((t	PROPN
ejpam-5587	544	13	−	−	PROPN
ejpam-5587	544	14	µi)|ker(eµ	µi)|ker(eµ	PROPN
ejpam-5587	544	15	)	)	PUNCT
ejpam-5587	544	16	)	)	PUNCT
ejpam-5587	545	1	=	=	SYM
ejpam-5587	545	2	ker(eµ	ker(eµ	NOUN
ejpam-5587	545	3	)	)	PUNCT
ejpam-5587	545	4	.	.	PUNCT
ejpam-5587	546	1	hence	hence	ADV
ejpam-5587	546	2	ran(t	ran(t	PROPN
ejpam-5587	546	3	−	−	PROPN
ejpam-5587	546	4	µi	µi	PROPN
ejpam-5587	546	5	)	)	PUNCT
ejpam-5587	546	6	is	be	AUX
ejpam-5587	546	7	closed	close	VERB
ejpam-5587	546	8	.	.	PUNCT
ejpam-5587	547	1	also	also	ADV
ejpam-5587	547	2	(	(	PUNCT
ejpam-5587	547	3	(	(	PUNCT
ejpam-5587	547	4	t	t	PROPN
ejpam-5587	547	5	−	−	PROPN
ejpam-5587	547	6	µi)|ker(eµ	µi)|ker(eµ	PROPN
ejpam-5587	547	7	)	)	PUNCT
ejpam-5587	547	8	)	)	PUNCT
ejpam-5587	547	9	−1	−1	NOUN
ejpam-5587	547	10	∈	∈	PROPN
ejpam-5587	547	11	b	b	PROPN
ejpam-5587	547	12	(	(	PUNCT
ejpam-5587	547	13	ker(eµ	ker(eµ	PROPN
ejpam-5587	547	14	)	)	PUNCT
ejpam-5587	547	15	)	)	PUNCT
ejpam-5587	547	16	.	.	PUNCT
ejpam-5587	548	1	thus	thus	ADV
ejpam-5587	548	2	we	we	PRON
ejpam-5587	548	3	get	get	VERB
ejpam-5587	548	4	dimker(t	dimker(t	NOUN
ejpam-5587	548	5	−	−	NOUN
ejpam-5587	548	6	µi)∗	µi)∗	PUNCT
ejpam-5587	548	7	=	=	SYM
ejpam-5587	548	8	dim	dim	NOUN
ejpam-5587	548	9	(	(	PUNCT
ejpam-5587	548	10	ran(t	ran(t	NOUN
ejpam-5587	548	11	−	−	PROPN
ejpam-5587	548	12	µi)⊥	µi)⊥	PROPN
ejpam-5587	548	13	)	)	PUNCT
ejpam-5587	548	14	=	=	PUNCT
ejpam-5587	549	1	dim	dim	ADJ
ejpam-5587	549	2	(	(	PUNCT
ejpam-5587	549	3	ker(eµ	ker(eµ	PROPN
ejpam-5587	549	4	)	)	PUNCT
ejpam-5587	549	5	⊥	⊥	NOUN
ejpam-5587	549	6	)	)	PUNCT
ejpam-5587	550	1	=	=	SYM
ejpam-5587	550	2	dim	dim	ADJ
ejpam-5587	550	3	(	(	PUNCT
ejpam-5587	550	4	ran(eµ	ran(eµ	NOUN
ejpam-5587	550	5	)	)	PUNCT
ejpam-5587	550	6	)	)	PUNCT
ejpam-5587	551	1	=	=	SYM
ejpam-5587	551	2	dim	dim	ADJ
ejpam-5587	551	3	(	(	PUNCT
ejpam-5587	551	4	ker(t	ker(t	NOUN
ejpam-5587	551	5	−	−	PROPN
ejpam-5587	551	6	µi	µi	PROPN
ejpam-5587	551	7	)	)	PUNCT
ejpam-5587	551	8	)	)	PUNCT
ejpam-5587	551	9	.	.	PUNCT
ejpam-5587	552	1	note	note	VERB
ejpam-5587	552	2	that	that	SCONJ
ejpam-5587	552	3	dim	dim	ADJ
ejpam-5587	552	4	(	(	PUNCT
ejpam-5587	552	5	ker(eµ	ker(eµ	NOUN
ejpam-5587	552	6	)	)	PUNCT
ejpam-5587	552	7	⊥	⊥	NOUN
ejpam-5587	552	8	)	)	PUNCT
ejpam-5587	552	9	=	=	SYM
ejpam-5587	552	10	dim	dim	ADJ
ejpam-5587	552	11	(	(	PUNCT
ejpam-5587	552	12	ran(eµ	ran(eµ	NOUN
ejpam-5587	552	13	)	)	PUNCT
ejpam-5587	552	14	)	)	PUNCT
ejpam-5587	553	1	but	but	CCONJ
ejpam-5587	553	2	the	the	DET
ejpam-5587	553	3	spaces	space	NOUN
ejpam-5587	553	4	,	,	PUNCT
ejpam-5587	553	5	ker(eµ	ker(eµ	NOUN
ejpam-5587	553	6	)	)	PUNCT
ejpam-5587	553	7	⊥	⊥	PROPN
ejpam-5587	553	8	and	and	CCONJ
ejpam-5587	553	9	ran(eµ	ran(eµ	NOUN
ejpam-5587	553	10	)	)	PUNCT
ejpam-5587	553	11	need	need	AUX
ejpam-5587	553	12	not	not	PART
ejpam-5587	553	13	be	be	AUX
ejpam-5587	553	14	the	the	DET
ejpam-5587	553	15	same	same	ADJ
ejpam-5587	553	16	.	.	PUNCT
ejpam-5587	554	1	hence	hence	ADV
ejpam-5587	554	2	t	t	PROPN
ejpam-5587	554	3	−µi	−µi	NOUN
ejpam-5587	554	4	is	be	AUX
ejpam-5587	554	5	fredholm	fredholm	NOUN
ejpam-5587	554	6	operator	operator	NOUN
ejpam-5587	554	7	of	of	ADP
ejpam-5587	554	8	index	index	NOUN
ejpam-5587	554	9	zero	zero	NUM
ejpam-5587	554	10	.	.	PUNCT
ejpam-5587	555	1	this	this	PRON
ejpam-5587	555	2	proves	prove	VERB
ejpam-5587	555	3	our	our	PRON
ejpam-5587	555	4	result	result	NOUN
ejpam-5587	555	5	.	.	PUNCT
ejpam-5587	556	1	theorem	theorem	ADJ
ejpam-5587	556	2	13	13	NUM
ejpam-5587	556	3	.	.	PUNCT
ejpam-5587	557	1	let	let	VERB
ejpam-5587	557	2	t	t	PROPN
ejpam-5587	557	3	∈	∈	PROPN
ejpam-5587	557	4	l(h	l(h	PROPN
ejpam-5587	557	5	)	)	PUNCT
ejpam-5587	557	6	be	be	AUX
ejpam-5587	557	7	a	a	DET
ejpam-5587	557	8	densely	densely	ADV
ejpam-5587	557	9	defined	define	VERB
ejpam-5587	557	10	closed	close	VERB
ejpam-5587	557	11	totally	totally	ADV
ejpam-5587	557	12	paranormal	paranormal	ADJ
ejpam-5587	557	13	operator	operator	NOUN
ejpam-5587	557	14	and	and	CCONJ
ejpam-5587	557	15	µ	µ	NOUN
ejpam-5587	557	16	be	be	AUX
ejpam-5587	557	17	a	a	DET
ejpam-5587	557	18	non	non	ADJ
ejpam-5587	557	19	-	-	ADJ
ejpam-5587	557	20	zero	zero	NUM
ejpam-5587	557	21	isolated	isolated	ADJ
ejpam-5587	557	22	point	point	NOUN
ejpam-5587	557	23	of	of	ADP
ejpam-5587	557	24	σ(t	σ(t	PROPN
ejpam-5587	557	25	)	)	PUNCT
ejpam-5587	557	26	.	.	PUNCT
ejpam-5587	558	1	then	then	ADV
ejpam-5587	558	2	the	the	DET
ejpam-5587	558	3	riesz	riesz	PROPN
ejpam-5587	558	4	projection	projection	NOUN
ejpam-5587	558	5	eµ	eµ	NOUN
ejpam-5587	558	6	with	with	ADP
ejpam-5587	558	7	respect	respect	NOUN
ejpam-5587	558	8	to	to	ADP
ejpam-5587	558	9	µ	µ	NUM
ejpam-5587	558	10	satisfy	satisfy	NOUN
ejpam-5587	558	11	ran(eµ	ran(eµ	NOUN
ejpam-5587	558	12	)	)	PUNCT
ejpam-5587	558	13	=	=	SYM
ejpam-5587	558	14	ker(t	ker(t	NOUN
ejpam-5587	558	15	−	−	NUM
ejpam-5587	558	16	µi	µi	NOUN
ejpam-5587	558	17	)	)	PUNCT
ejpam-5587	558	18	=	=	PUNCT
ejpam-5587	558	19	ker(t	ker(t	NOUN
ejpam-5587	558	20	−	−	NOUN
ejpam-5587	558	21	µi)∗.	µi)∗.	ADV
ejpam-5587	558	22	moreover	moreover	ADV
ejpam-5587	558	23	,	,	PUNCT
ejpam-5587	558	24	eµ	eµ	PROPN
ejpam-5587	558	25	is	be	AUX
ejpam-5587	558	26	self	self	NOUN
ejpam-5587	558	27	-	-	PUNCT
ejpam-5587	558	28	adjoint	adjoint	NOUN
ejpam-5587	558	29	.	.	PUNCT
ejpam-5587	559	1	proof	proof	NOUN
ejpam-5587	559	2	.	.	PUNCT
ejpam-5587	560	1	let	let	VERB
ejpam-5587	560	2	µ	µ	X
ejpam-5587	560	3	be	be	AUX
ejpam-5587	560	4	a	a	DET
ejpam-5587	560	5	non	non	ADJ
ejpam-5587	560	6	-	-	ADJ
ejpam-5587	560	7	zero	zero	NUM
ejpam-5587	560	8	isolated	isolated	ADJ
ejpam-5587	560	9	point	point	NOUN
ejpam-5587	560	10	of	of	ADP
ejpam-5587	560	11	σ(t	σ(t	PROPN
ejpam-5587	560	12	)	)	PUNCT
ejpam-5587	560	13	.	.	PUNCT
ejpam-5587	561	1	by	by	ADP
ejpam-5587	561	2	theorem	theorem	NOUN
ejpam-5587	561	3	1	1	NUM
ejpam-5587	561	4	and	and	CCONJ
ejpam-5587	561	5	theorem	theorem	VERB
ejpam-5587	561	6	7	7	NUM
ejpam-5587	561	7	,	,	PUNCT
ejpam-5587	561	8	µ	µ	X
ejpam-5587	561	9	/∈	/∈	PUNCT
ejpam-5587	561	10	σ	σ	PROPN
ejpam-5587	561	11	(	(	PUNCT
ejpam-5587	561	12	t	t	PROPN
ejpam-5587	561	13	|ker(eµ	|ker(eµ	PROPN
ejpam-5587	561	14	)	)	PUNCT
ejpam-5587	561	15	)	)	PUNCT
ejpam-5587	562	1	and	and	CCONJ
ejpam-5587	562	2	ran(t	ran(t	PROPN
ejpam-5587	562	3	−µi	−µi	NOUN
ejpam-5587	562	4	)	)	PUNCT
ejpam-5587	563	1	=	=	SYM
ejpam-5587	563	2	ker(eµ	ker(eµ	NOUN
ejpam-5587	563	3	)	)	PUNCT
ejpam-5587	563	4	.	.	PUNCT
ejpam-5587	564	1	that	that	PRON
ejpam-5587	564	2	means	mean	VERB
ejpam-5587	564	3	(	(	PUNCT
ejpam-5587	564	4	t	t	PROPN
ejpam-5587	564	5	−µi)|ker(eµ	−µi)|ker(eµ	PROPN
ejpam-5587	564	6	)	)	PUNCT
ejpam-5587	564	7	:	:	PUNCT
ejpam-5587	564	8	ker(eµ)∩d(t	ker(eµ)∩d(t	NOUN
ejpam-5587	564	9	)	)	PUNCT
ejpam-5587	564	10	→	→	SYM
ejpam-5587	564	11	ker(eµ	ker(eµ	X
ejpam-5587	564	12	)	)	PUNCT
ejpam-5587	564	13	=	=	SYM
ejpam-5587	564	14	ran(t−µi	ran(t−µi	NOUN
ejpam-5587	564	15	)	)	PUNCT
ejpam-5587	564	16	is	be	AUX
ejpam-5587	564	17	a	a	DET
ejpam-5587	564	18	bijection	bijection	NOUN
ejpam-5587	564	19	.	.	PUNCT
ejpam-5587	565	1	also	also	ADV
ejpam-5587	565	2	(	(	PUNCT
ejpam-5587	565	3	t−µi)|ker(t−µi)⊥∩d(t	t−µi)|ker(t−µi)⊥∩d(t	ADJ
ejpam-5587	565	4	)	)	PUNCT
ejpam-5587	566	1	:	:	PUNCT
ejpam-5587	566	2	ker(t−µi)∩d(t	ker(t−µi)∩d(t	ADJ
ejpam-5587	566	3	)	)	PUNCT
ejpam-5587	566	4	→	→	SYM
ejpam-5587	567	1	ran(t	ran(t	X
ejpam-5587	567	2	−	−	NOUN
ejpam-5587	567	3	µi	µi	PROPN
ejpam-5587	567	4	)	)	PUNCT
ejpam-5587	567	5	is	be	AUX
ejpam-5587	567	6	a	a	DET
ejpam-5587	567	7	bijection	bijection	NOUN
ejpam-5587	567	8	,	,	PUNCT
ejpam-5587	567	9	we	we	PRON
ejpam-5587	567	10	have	have	VERB
ejpam-5587	567	11	ker(eµ	ker(eµ	NOUN
ejpam-5587	567	12	)	)	PUNCT
ejpam-5587	568	1	∩d(t	∩d(t	ADJ
ejpam-5587	568	2	)	)	PUNCT
ejpam-5587	568	3	⊆	⊆	NUM
ejpam-5587	568	4	ker(t	ker(t	NOUN
ejpam-5587	568	5	−	−	NOUN
ejpam-5587	568	6	µi)⊥	µi)⊥	PROPN
ejpam-5587	568	7	∩d(t	∩d(t	ADJ
ejpam-5587	568	8	)	)	PUNCT
ejpam-5587	568	9	.	.	PUNCT
ejpam-5587	569	1	now	now	ADV
ejpam-5587	569	2	we	we	PRON
ejpam-5587	569	3	claim	claim	VERB
ejpam-5587	569	4	that	that	SCONJ
ejpam-5587	569	5	ker(eµ)∩d(t	ker(eµ)∩d(t	NOUN
ejpam-5587	569	6	)	)	PUNCT
ejpam-5587	570	1	=	=	SYM
ejpam-5587	570	2	ker(t	ker(t	NOUN
ejpam-5587	570	3	−µi)⊥	−µi)⊥	NOUN
ejpam-5587	570	4	∩d(t	∩d(t	ADJ
ejpam-5587	570	5	)	)	PUNCT
ejpam-5587	570	6	.	.	PUNCT
ejpam-5587	571	1	let	let	VERB
ejpam-5587	571	2	x	x	PUNCT
ejpam-5587	571	3	∈	∈	NOUN
ejpam-5587	571	4	ker(t	ker(t	NOUN
ejpam-5587	571	5	−µi)⊥	−µi)⊥	NOUN
ejpam-5587	571	6	∩d(t	∩d(t	ADJ
ejpam-5587	571	7	)	)	PUNCT
ejpam-5587	571	8	and	and	CCONJ
ejpam-5587	571	9	eµx	eµx	NOUN
ejpam-5587	571	10	=	=	SYM
ejpam-5587	571	11	p+	p+	NOUN
ejpam-5587	571	12	q	q	X
ejpam-5587	571	13	,	,	PUNCT
ejpam-5587	571	14	where	where	SCONJ
ejpam-5587	571	15	p	p	PROPN
ejpam-5587	571	16	∈	∈	PROPN
ejpam-5587	571	17	ker(t	ker(t	NOUN
ejpam-5587	571	18	−	−	PROPN
ejpam-5587	571	19	µi	µi	PROPN
ejpam-5587	571	20	)	)	PUNCT
ejpam-5587	571	21	,	,	PUNCT
ejpam-5587	571	22	q	q	PROPN
ejpam-5587	571	23	∈	∈	PROPN
ejpam-5587	571	24	ker(t	ker(t	NOUN
ejpam-5587	571	25	−	−	NOUN
ejpam-5587	571	26	µi)⊥.	µi)⊥.	DET
ejpam-5587	571	27	operating	operate	VERB
ejpam-5587	571	28	eµ	eµ	NOUN
ejpam-5587	571	29	on	on	ADP
ejpam-5587	571	30	both	both	DET
ejpam-5587	571	31	sides	side	NOUN
ejpam-5587	571	32	,	,	PUNCT
ejpam-5587	571	33	we	we	PRON
ejpam-5587	571	34	get	get	VERB
ejpam-5587	571	35	p+	p+	ADV
ejpam-5587	571	36	q	q	NOUN
ejpam-5587	571	37	=	=	PUNCT
ejpam-5587	571	38	eµx	eµx	NOUN
ejpam-5587	571	39	=	=	SYM
ejpam-5587	571	40	p+	p+	NOUN
ejpam-5587	571	41	eµq	eµq	NOUN
ejpam-5587	571	42	.	.	PUNCT
ejpam-5587	572	1	this	this	PRON
ejpam-5587	572	2	implies	imply	VERB
ejpam-5587	572	3	eµq	eµq	NOUN
ejpam-5587	572	4	=	=	SYM
ejpam-5587	572	5	q	q	PROPN
ejpam-5587	572	6	∈	∈	PROPN
ejpam-5587	572	7	ran(eµ	ran(eµ	NOUN
ejpam-5587	572	8	)	)	PUNCT
ejpam-5587	572	9	∩	∩	ADJ
ejpam-5587	572	10	ker(t	ker(t	NOUN
ejpam-5587	572	11	−	−	NOUN
ejpam-5587	572	12	µi)⊥	µi)⊥	PROPN
ejpam-5587	572	13	=	=	SYM
ejpam-5587	572	14	{	{	PUNCT
ejpam-5587	572	15	0	0	NUM
ejpam-5587	572	16	}	}	PUNCT
ejpam-5587	572	17	,	,	PUNCT
ejpam-5587	572	18	by	by	ADP
ejpam-5587	572	19	theorem	theorem	NOUN
ejpam-5587	572	20	6	6	NUM
ejpam-5587	572	21	.	.	PUNCT
ejpam-5587	572	22	from	from	ADP
ejpam-5587	572	23	this	this	PRON
ejpam-5587	572	24	we	we	PRON
ejpam-5587	572	25	conclude	conclude	VERB
ejpam-5587	572	26	that	that	PRON
ejpam-5587	572	27	eµx	eµx	NOUN
ejpam-5587	573	1	=	=	PUNCT
ejpam-5587	573	2	p	p	X
ejpam-5587	573	3	=	=	NOUN
ejpam-5587	573	4	eµp	eµp	NOUN
ejpam-5587	573	5	,	,	PUNCT
ejpam-5587	573	6	that	that	ADV
ejpam-5587	573	7	is	is	ADV
ejpam-5587	573	8	,	,	PUNCT
ejpam-5587	573	9	x−p	x−p	PROPN
ejpam-5587	573	10	∈	∈	PROPN
ejpam-5587	573	11	ker(eµ)∩d(t	ker(eµ)∩d(t	NOUN
ejpam-5587	573	12	)	)	PUNCT
ejpam-5587	574	1	⊆	⊆	NUM
ejpam-5587	574	2	ker(t	ker(t	NOUN
ejpam-5587	574	3	−µi)⊥∩d(t	−µi)⊥∩d(t	NOUN
ejpam-5587	574	4	)	)	PUNCT
ejpam-5587	574	5	.	.	PUNCT
ejpam-5587	575	1	as	as	ADP
ejpam-5587	575	2	x	x	PROPN
ejpam-5587	575	3	∈	∈	PROPN
ejpam-5587	575	4	ker(t	ker(t	NOUN
ejpam-5587	575	5	−	−	NOUN
ejpam-5587	575	6	µi)⊥	µi)⊥	PROPN
ejpam-5587	575	7	,	,	PUNCT
ejpam-5587	575	8	we	we	PRON
ejpam-5587	575	9	get	get	VERB
ejpam-5587	575	10	p	p	NOUN
ejpam-5587	575	11	∈	∈	NOUN
ejpam-5587	575	12	ker(t	ker(t	NOUN
ejpam-5587	575	13	−	−	PROPN
ejpam-5587	575	14	µi	µi	NOUN
ejpam-5587	575	15	)	)	PUNCT
ejpam-5587	575	16	∩	∩	ADJ
ejpam-5587	575	17	ker(t	ker(t	NOUN
ejpam-5587	575	18	−	−	NOUN
ejpam-5587	575	19	µi)⊥	µi)⊥	PROPN
ejpam-5587	575	20	=	=	SYM
ejpam-5587	575	21	{	{	PUNCT
ejpam-5587	575	22	0	0	NUM
ejpam-5587	575	23	}	}	PUNCT
ejpam-5587	575	24	.	.	PUNCT
ejpam-5587	576	1	consequently	consequently	ADV
ejpam-5587	576	2	eµx	eµx	VERB
ejpam-5587	576	3	=	=	SYM
ejpam-5587	576	4	0	0	X
ejpam-5587	576	5	.	.	PUNCT
ejpam-5587	577	1	so	so	ADV
ejpam-5587	577	2	ker(t	ker(t	NOUN
ejpam-5587	577	3	−µi)⊥∩d(t	−µi)⊥∩d(t	NOUN
ejpam-5587	577	4	)	)	PUNCT
ejpam-5587	577	5	⊆	⊆	NUM
ejpam-5587	577	6	ker(eµ)∩d(t	ker(eµ)∩d(t	NOUN
ejpam-5587	577	7	)	)	PUNCT
ejpam-5587	577	8	.	.	PUNCT
ejpam-5587	578	1	hence	hence	ADV
ejpam-5587	578	2	ker(t	ker(t	NOUN
ejpam-5587	578	3	−µi)⊥∩d(t	−µi)⊥∩d(t	NOUN
ejpam-5587	578	4	)	)	PUNCT
ejpam-5587	579	1	=	=	SYM
ejpam-5587	579	2	ker(eµ)∩d(t	ker(eµ)∩d(t	NOUN
ejpam-5587	579	3	)	)	PUNCT
ejpam-5587	579	4	.	.	PUNCT
ejpam-5587	580	1	by	by	ADP
ejpam-5587	580	2	lemma	lemma	PROPN
ejpam-5587	580	3	1	1	NUM
ejpam-5587	580	4	and	and	CCONJ
ejpam-5587	580	5	theorem	theorem	VERB
ejpam-5587	580	6	7	7	NUM
ejpam-5587	580	7	,	,	PUNCT
ejpam-5587	580	8	we	we	PRON
ejpam-5587	580	9	get	get	VERB
ejpam-5587	580	10	ker(t	ker(t	NOUN
ejpam-5587	580	11	−	−	NOUN
ejpam-5587	581	1	µi)⊥	µi)⊥	PROPN
ejpam-5587	581	2	=	=	SYM
ejpam-5587	581	3	ker(t	ker(t	NOUN
ejpam-5587	581	4	−	−	PROPN
ejpam-5587	581	5	µi)⊥	µi)⊥	PROPN
ejpam-5587	581	6	∩d(t	∩d(t	ADJ
ejpam-5587	581	7	)	)	PUNCT
ejpam-5587	581	8	=	=	SYM
ejpam-5587	581	9	ker(eµ	ker(eµ	ADJ
ejpam-5587	581	10	)	)	PUNCT
ejpam-5587	581	11	∩d(t	∩d(t	ADJ
ejpam-5587	581	12	)	)	PUNCT
ejpam-5587	581	13	=	=	SYM
ejpam-5587	582	1	ran(t	ran(t	ADJ
ejpam-5587	582	2	−	−	PROPN
ejpam-5587	582	3	µi	µi	PROPN
ejpam-5587	582	4	)	)	PUNCT
ejpam-5587	582	5	∩d(t	∩d(t	ADJ
ejpam-5587	582	6	)	)	PUNCT
ejpam-5587	582	7	=	=	PUNCT
ejpam-5587	583	1	(	(	PUNCT
ejpam-5587	583	2	ker(t	ker(t	NOUN
ejpam-5587	583	3	−	−	PROPN
ejpam-5587	583	4	µi)∗)⊥	µi)∗)⊥	ADJ
ejpam-5587	583	5	∩d(t	∩d(t	ADJ
ejpam-5587	583	6	)	)	PUNCT
ejpam-5587	583	7	⊆	⊆	NUM
ejpam-5587	583	8	(	(	PUNCT
ejpam-5587	583	9	ker(t	ker(t	NOUN
ejpam-5587	583	10	−	−	NOUN
ejpam-5587	583	11	µi)∗)⊥.	µi)∗)⊥.	NOUN
ejpam-5587	583	12	hence	hence	ADV
ejpam-5587	583	13	ker(t	ker(t	NOUN
ejpam-5587	583	14	−	−	NOUN
ejpam-5587	583	15	µi)∗	µi)∗	NUM
ejpam-5587	583	16	⊆	⊆	NUM
ejpam-5587	583	17	ker(t	ker(t	NOUN
ejpam-5587	583	18	−	−	PROPN
ejpam-5587	583	19	µi	µi	PROPN
ejpam-5587	583	20	)	)	PUNCT
ejpam-5587	583	21	.	.	PUNCT
ejpam-5587	584	1	by	by	ADP
ejpam-5587	584	2	theorem	theorem	ADJ
ejpam-5587	584	3	7	7	NUM
ejpam-5587	584	4	,	,	PUNCT
ejpam-5587	584	5	ker(eµ	ker(eµ	NOUN
ejpam-5587	584	6	)	)	PUNCT
ejpam-5587	585	1	⊥	⊥	NOUN
ejpam-5587	585	2	=	=	PUNCT
ejpam-5587	585	3	ran(t	ran(t	ADJ
ejpam-5587	585	4	−	−	NOUN
ejpam-5587	585	5	µi)⊥	µi)⊥	PROPN
ejpam-5587	585	6	=	=	PUNCT
ejpam-5587	585	7	ker((t	ker((t	VERB
ejpam-5587	585	8	−	−	NOUN
ejpam-5587	585	9	µi)∗	µi)∗	NUM
ejpam-5587	585	10	)	)	PUNCT
ejpam-5587	585	11	⊆	⊆	NUM
ejpam-5587	585	12	ker(t	ker(t	NOUN
ejpam-5587	585	13	−	−	PROPN
ejpam-5587	585	14	µi	µi	PROPN
ejpam-5587	585	15	)	)	PUNCT
ejpam-5587	585	16	=	=	SYM
ejpam-5587	585	17	ran(eµ	ran(eµ	NOUN
ejpam-5587	585	18	)	)	PUNCT
ejpam-5587	585	19	.	.	PUNCT
ejpam-5587	586	1	hence	hence	ADV
ejpam-5587	586	2	ker(eµ	ker(eµ	X
ejpam-5587	586	3	)	)	PUNCT
ejpam-5587	587	1	⊥	⊥	NOUN
ejpam-5587	587	2	=	=	PUNCT
ejpam-5587	587	3	ran(t	ran(t	ADJ
ejpam-5587	587	4	−	−	NOUN
ejpam-5587	587	5	µi)⊥	µi)⊥	PROPN
ejpam-5587	587	6	=	=	PUNCT
ejpam-5587	587	7	ker((t	ker((t	VERB
ejpam-5587	587	8	−	−	NOUN
ejpam-5587	587	9	µi)∗	µi)∗	NUM
ejpam-5587	587	10	)	)	PUNCT
ejpam-5587	587	11	⊆	⊆	NUM
ejpam-5587	587	12	ker(t	ker(t	NOUN
ejpam-5587	587	13	−	−	PROPN
ejpam-5587	587	14	µi	µi	PROPN
ejpam-5587	587	15	)	)	PUNCT
ejpam-5587	587	16	=	=	SYM
ejpam-5587	587	17	ran(eµ	ran(eµ	NOUN
ejpam-5587	587	18	)	)	PUNCT
ejpam-5587	587	19	.	.	PUNCT
ejpam-5587	588	1	hence	hence	ADV
ejpam-5587	588	2	ker(eµ	ker(eµ	PROPN
ejpam-5587	588	3	)	)	PUNCT
ejpam-5587	589	1	⊥	⊥	ADJ
ejpam-5587	589	2	⊆	⊆	NUM
ejpam-5587	589	3	ran(eµ	ran(eµ	NOUN
ejpam-5587	589	4	)	)	PUNCT
ejpam-5587	589	5	.	.	PUNCT
ejpam-5587	590	1	if	if	SCONJ
ejpam-5587	590	2	x	x	SYM
ejpam-5587	590	3	∈	∈	PROPN
ejpam-5587	590	4	ran(eµ	ran(eµ	NOUN
ejpam-5587	590	5	)	)	PUNCT
ejpam-5587	590	6	,	,	PUNCT
ejpam-5587	590	7	then	then	ADV
ejpam-5587	590	8	x	x	X
ejpam-5587	590	9	=	=	PUNCT
ejpam-5587	590	10	u	u	NOUN
ejpam-5587	590	11	+	+	NOUN
ejpam-5587	590	12	v	v	ADP
ejpam-5587	590	13	where	where	SCONJ
ejpam-5587	590	14	u	u	NOUN
ejpam-5587	590	15	∈	∈	PROPN
ejpam-5587	590	16	ker(eµ	ker(eµ	PROPN
ejpam-5587	590	17	)	)	PUNCT
ejpam-5587	590	18	and	and	CCONJ
ejpam-5587	590	19	v	v	ADP
ejpam-5587	590	20	∈	∈	PROPN
ejpam-5587	590	21	ker(eµ	ker(eµ	NOUN
ejpam-5587	590	22	)	)	PUNCT
ejpam-5587	590	23	⊥.	⊥.	NUM
ejpam-5587	590	24	as	as	ADP
ejpam-5587	590	25	ker(eµ	ker(eµ	PROPN
ejpam-5587	590	26	)	)	PUNCT
ejpam-5587	590	27	⊥	⊥	PROPN
ejpam-5587	590	28	⊆	⊆	NUM
ejpam-5587	590	29	ran(eµ	ran(eµ	NOUN
ejpam-5587	590	30	)	)	PUNCT
ejpam-5587	590	31	,	,	PUNCT
ejpam-5587	590	32	we	we	PRON
ejpam-5587	590	33	get	get	VERB
ejpam-5587	590	34	u	u	NOUN
ejpam-5587	590	35	=	=	NOUN
ejpam-5587	590	36	x	x	SYM
ejpam-5587	590	37	−	−	PROPN
ejpam-5587	590	38	v	v	NUM
ejpam-5587	590	39	∈	∈	PROPN
ejpam-5587	590	40	ker(eµ	ker(eµ	NOUN
ejpam-5587	590	41	)	)	PUNCT
ejpam-5587	590	42	∩	∩	ADJ
ejpam-5587	590	43	ran(eµ	ran(eµ	NOUN
ejpam-5587	590	44	)	)	PUNCT
ejpam-5587	590	45	=	=	PUNCT
ejpam-5587	590	46	{	{	PUNCT
ejpam-5587	590	47	0	0	NUM
ejpam-5587	590	48	}	}	PUNCT
ejpam-5587	590	49	.	.	PUNCT
ejpam-5587	591	1	thus	thus	ADV
ejpam-5587	591	2	we	we	PRON
ejpam-5587	591	3	get	get	VERB
ejpam-5587	591	4	ker(eµ	ker(eµ	NOUN
ejpam-5587	591	5	)	)	PUNCT
ejpam-5587	592	1	⊥	⊥	NOUN
ejpam-5587	592	2	=	=	SYM
ejpam-5587	592	3	ran(eµ	ran(eµ	NOUN
ejpam-5587	592	4	)	)	PUNCT
ejpam-5587	592	5	,	,	PUNCT
ejpam-5587	592	6	which	which	PRON
ejpam-5587	592	7	is	be	AUX
ejpam-5587	592	8	equivalent	equivalent	ADJ
ejpam-5587	592	9	to	to	PART
ejpam-5587	592	10	say	say	VERB
ejpam-5587	592	11	that	that	SCONJ
ejpam-5587	592	12	ker(t−µi	ker(t−µi	NOUN
ejpam-5587	592	13	)	)	PUNCT
ejpam-5587	593	1	=	=	PUNCT
ejpam-5587	593	2	ker(t−µi)∗.	ker(t−µi)∗.	PROPN
ejpam-5587	593	3	as	as	ADP
ejpam-5587	593	4	ker(eµ	ker(eµ	PROPN
ejpam-5587	593	5	)	)	PUNCT
ejpam-5587	593	6	⊥	⊥	NOUN
ejpam-5587	593	7	=	=	SYM
ejpam-5587	593	8	ran(eµ	ran(eµ	NOUN
ejpam-5587	593	9	)	)	PUNCT
ejpam-5587	593	10	,	,	PUNCT
ejpam-5587	593	11	we	we	PRON
ejpam-5587	593	12	have	have	VERB
ejpam-5587	593	13	that	that	DET
ejpam-5587	593	14	eµ	eµ	NOUN
ejpam-5587	593	15	is	be	AUX
ejpam-5587	593	16	an	an	DET
ejpam-5587	593	17	orthogonal	orthogonal	ADJ
ejpam-5587	593	18	projection	projection	NOUN
ejpam-5587	593	19	.	.	PUNCT
ejpam-5587	594	1	hence	hence	ADV
ejpam-5587	594	2	eµ	eµ	PROPN
ejpam-5587	594	3	is	be	AUX
ejpam-5587	594	4	self	self	NOUN
ejpam-5587	594	5	-	-	PUNCT
ejpam-5587	594	6	adjoint	adjoint	NOUN
ejpam-5587	594	7	.	.	PUNCT
ejpam-5587	595	1	from	from	ADP
ejpam-5587	595	2	the	the	DET
ejpam-5587	595	3	proof	proof	NOUN
ejpam-5587	595	4	of	of	ADP
ejpam-5587	595	5	theorem	theorem	ADJ
ejpam-5587	595	6	13	13	NUM
ejpam-5587	595	7	,	,	PUNCT
ejpam-5587	595	8	we	we	PRON
ejpam-5587	595	9	have	have	VERB
ejpam-5587	595	10	s.	s.	PROPN
ejpam-5587	595	11	alnabulsi	alnabulsi	PROPN
ejpam-5587	595	12	,	,	PUNCT
ejpam-5587	595	13	m.h.m	m.h.m	PROPN
ejpam-5587	595	14	.	.	PUNCT
ejpam-5587	595	15	rashid	rashid	PROPN
ejpam-5587	595	16	/	/	SYM
ejpam-5587	595	17	eur	eur	PROPN
ejpam-5587	595	18	.	.	PUNCT
ejpam-5587	596	1	j.	j.	PROPN
ejpam-5587	596	2	pure	pure	PROPN
ejpam-5587	596	3	appl	appl	PROPN
ejpam-5587	596	4	.	.	PROPN
ejpam-5587	596	5	math	math	PROPN
ejpam-5587	596	6	,	,	PUNCT
ejpam-5587	596	7	18	18	NUM
ejpam-5587	596	8	(	(	PUNCT
ejpam-5587	596	9	1	1	NUM
ejpam-5587	596	10	)	)	PUNCT
ejpam-5587	596	11	(	(	PUNCT
ejpam-5587	596	12	2025	2025	NUM
ejpam-5587	596	13	)	)	PUNCT
ejpam-5587	596	14	,	,	PUNCT
ejpam-5587	596	15	5587	5587	NUM
ejpam-5587	596	16	16	16	NUM
ejpam-5587	596	17	of	of	ADP
ejpam-5587	596	18	20	20	NUM
ejpam-5587	596	19	corollary	corollary	ADJ
ejpam-5587	596	20	3	3	NUM
ejpam-5587	596	21	.	.	PUNCT
ejpam-5587	597	1	let	let	VERB
ejpam-5587	597	2	t	t	PROPN
ejpam-5587	597	3	∈	∈	PROPN
ejpam-5587	597	4	l(h	l(h	PROPN
ejpam-5587	597	5	)	)	PUNCT
ejpam-5587	597	6	be	be	AUX
ejpam-5587	597	7	a	a	DET
ejpam-5587	597	8	densely	densely	ADV
ejpam-5587	597	9	defined	define	VERB
ejpam-5587	597	10	closed	closed	ADJ
ejpam-5587	597	11	paranormal	paranormal	ADJ
ejpam-5587	597	12	operator	operator	NOUN
ejpam-5587	597	13	.	.	PUNCT
ejpam-5587	598	1	then	then	ADV
ejpam-5587	598	2	ker(t	ker(t	NOUN
ejpam-5587	598	3	−	−	PROPN
ejpam-5587	598	4	µi)∗	µi)∗	PUNCT
ejpam-5587	598	5	⊂	⊂	PROPN
ejpam-5587	599	1	ker(t	ker(t	PROPN
ejpam-5587	599	2	−	−	PROPN
ejpam-5587	599	3	µi	µi	PROPN
ejpam-5587	599	4	)	)	PUNCT
ejpam-5587	599	5	for	for	ADP
ejpam-5587	599	6	all	all	DET
ejpam-5587	599	7	µ	µ	PRON
ejpam-5587	599	8	∈	∈	PROPN
ejpam-5587	599	9	c.	c.	NOUN
ejpam-5587	599	10	applying	apply	VERB
ejpam-5587	599	11	the	the	DET
ejpam-5587	599	12	concept	concept	NOUN
ejpam-5587	599	13	of	of	ADP
ejpam-5587	599	14	birkhoff	birkhoff	NOUN
ejpam-5587	599	15	–	–	PUNCT
ejpam-5587	599	16	james	james	PROPN
ejpam-5587	599	17	orthogonality	orthogonality	PROPN
ejpam-5587	599	18	,	,	PUNCT
ejpam-5587	599	19	we	we	PRON
ejpam-5587	599	20	show	show	VERB
ejpam-5587	599	21	that	that	SCONJ
ejpam-5587	599	22	for	for	ADP
ejpam-5587	599	23	a	a	DET
ejpam-5587	599	24	paranormal	paranormal	ADJ
ejpam-5587	599	25	operator	operator	NOUN
ejpam-5587	599	26	,	,	PUNCT
ejpam-5587	599	27	the	the	DET
ejpam-5587	599	28	eigenspaces	eigenspace	NOUN
ejpam-5587	599	29	corresponding	correspond	VERB
ejpam-5587	599	30	to	to	ADP
ejpam-5587	599	31	distinct	distinct	ADJ
ejpam-5587	599	32	isolated	isolated	ADJ
ejpam-5587	599	33	eigenvalues	eigenvalue	NOUN
ejpam-5587	599	34	are	be	AUX
ejpam-5587	599	35	entirely	entirely	ADV
ejpam-5587	599	36	independent	independent	ADJ
ejpam-5587	599	37	of	of	ADP
ejpam-5587	599	38	one	one	NUM
ejpam-5587	599	39	another	another	DET
ejpam-5587	599	40	.	.	PUNCT
ejpam-5587	600	1	specifically	specifically	ADV
ejpam-5587	600	2	,	,	PUNCT
ejpam-5587	600	3	let	let	VERB
ejpam-5587	600	4	m	m	PRON
ejpam-5587	600	5	be	be	AUX
ejpam-5587	600	6	a	a	DET
ejpam-5587	600	7	subspace	subspace	NOUN
ejpam-5587	600	8	within	within	ADP
ejpam-5587	600	9	a	a	DET
ejpam-5587	600	10	banach	banach	NOUN
ejpam-5587	600	11	space	space	NOUN
ejpam-5587	600	12	x.	x.	NOUN
ejpam-5587	601	1	we	we	PRON
ejpam-5587	601	2	say	say	VERB
ejpam-5587	601	3	that	that	SCONJ
ejpam-5587	601	4	m	m	PROPN
ejpam-5587	601	5	is	be	AUX
ejpam-5587	601	6	birkhoff	birkhoff	NOUN
ejpam-5587	601	7	–	–	PUNCT
ejpam-5587	601	8	james	james	PROPN
ejpam-5587	601	9	orthogonal	orthogonal	PROPN
ejpam-5587	601	10	to	to	ADP
ejpam-5587	601	11	another	another	DET
ejpam-5587	601	12	subspace	subspace	NOUN
ejpam-5587	601	13	n	n	CCONJ
ejpam-5587	601	14	in	in	ADP
ejpam-5587	601	15	x	x	PUNCT
ejpam-5587	601	16	if	if	SCONJ
ejpam-5587	601	17	∥m∥	∥m∥	VERB
ejpam-5587	601	18	≤	≤	PUNCT
ejpam-5587	601	19	∥m+	∥m+	X
ejpam-5587	601	20	n∥	n∥	NOUN
ejpam-5587	601	21	for	for	ADP
ejpam-5587	601	22	all	all	DET
ejpam-5587	601	23	m	m	NOUN
ejpam-5587	601	24	∈	∈	NOUN
ejpam-5587	601	25	m	m	NOUN
ejpam-5587	601	26	and	and	CCONJ
ejpam-5587	601	27	n	n	PRON
ejpam-5587	601	28	∈	∈	PROPN
ejpam-5587	601	29	n	n	NOUN
ejpam-5587	601	30	.	.	PUNCT
ejpam-5587	602	1	this	this	DET
ejpam-5587	602	2	definition	definition	NOUN
ejpam-5587	602	3	aligns	align	VERB
ejpam-5587	602	4	with	with	ADP
ejpam-5587	602	5	the	the	DET
ejpam-5587	602	6	concept	concept	NOUN
ejpam-5587	602	7	of	of	ADP
ejpam-5587	602	8	birkhoff	birkhoff	NOUN
ejpam-5587	602	9	–	–	PUNCT
ejpam-5587	602	10	james	james	PROPN
ejpam-5587	602	11	orthogonality	orthogonality	NOUN
ejpam-5587	602	12	,	,	PUNCT
ejpam-5587	602	13	which	which	PRON
ejpam-5587	602	14	,	,	PUNCT
ejpam-5587	602	15	in	in	ADP
ejpam-5587	602	16	the	the	DET
ejpam-5587	602	17	case	case	NOUN
ejpam-5587	602	18	of	of	ADP
ejpam-5587	602	19	a	a	DET
ejpam-5587	602	20	hilbert	hilbert	NOUN
ejpam-5587	602	21	space	space	NOUN
ejpam-5587	602	22	x	x	NOUN
ejpam-5587	602	23	,	,	PUNCT
ejpam-5587	602	24	matches	match	VERB
ejpam-5587	602	25	the	the	DET
ejpam-5587	602	26	traditional	traditional	ADJ
ejpam-5587	602	27	notion	notion	NOUN
ejpam-5587	602	28	of	of	ADP
ejpam-5587	602	29	orthogonality	orthogonality	NOUN
ejpam-5587	602	30	.	.	PUNCT
ejpam-5587	603	1	proposition	proposition	NOUN
ejpam-5587	603	2	6	6	NUM
ejpam-5587	603	3	.	.	PUNCT
ejpam-5587	604	1	let	let	VERB
ejpam-5587	604	2	t	t	PROPN
ejpam-5587	604	3	∈	∈	PROPN
ejpam-5587	604	4	l(h	l(h	PROPN
ejpam-5587	604	5	)	)	PUNCT
ejpam-5587	604	6	be	be	AUX
ejpam-5587	604	7	a	a	DET
ejpam-5587	604	8	densely	densely	ADV
ejpam-5587	604	9	defined	define	VERB
ejpam-5587	604	10	closed	close	VERB
ejpam-5587	604	11	totally	totally	ADV
ejpam-5587	604	12	paranormal	paranormal	ADJ
ejpam-5587	604	13	operator	operator	NOUN
ejpam-5587	604	14	.	.	PUNCT
ejpam-5587	605	1	if	if	SCONJ
ejpam-5587	605	2	µ1	µ1	PROPN
ejpam-5587	605	3	and	and	CCONJ
ejpam-5587	605	4	µ2	µ2	PROPN
ejpam-5587	605	5	are	be	AUX
ejpam-5587	605	6	two	two	NUM
ejpam-5587	605	7	non	non	ADJ
ejpam-5587	605	8	zero	zero	NUM
ejpam-5587	605	9	distinct	distinct	ADJ
ejpam-5587	605	10	isolated	isolate	VERB
ejpam-5587	605	11	points	point	NOUN
ejpam-5587	605	12	of	of	ADP
ejpam-5587	605	13	σ(t	σ(t	PROPN
ejpam-5587	605	14	)	)	PUNCT
ejpam-5587	605	15	,	,	PUNCT
ejpam-5587	605	16	then	then	ADV
ejpam-5587	605	17	ker(t−µ1i	ker(t−µ1i	PROPN
ejpam-5587	605	18	)	)	PUNCT
ejpam-5587	605	19	is	be	AUX
ejpam-5587	605	20	orthogonal	orthogonal	ADJ
ejpam-5587	605	21	to	to	ADP
ejpam-5587	605	22	ker(t	ker(t	PROPN
ejpam-5587	605	23	−	−	PROPN
ejpam-5587	605	24	µ2i	µ2i	ADJ
ejpam-5587	605	25	)	)	PUNCT
ejpam-5587	605	26	.	.	PUNCT
ejpam-5587	606	1	proof	proof	NOUN
ejpam-5587	606	2	.	.	PUNCT
ejpam-5587	607	1	without	without	ADP
ejpam-5587	607	2	loss	loss	NOUN
ejpam-5587	607	3	of	of	ADP
ejpam-5587	607	4	generality	generality	NOUN
ejpam-5587	607	5	,	,	PUNCT
ejpam-5587	607	6	assume	assume	VERB
ejpam-5587	607	7	that	that	SCONJ
ejpam-5587	607	8	|µ1|	|µ1|	NOUN
ejpam-5587	607	9	<	<	X
ejpam-5587	607	10	|µ2|	|µ2|	NOUN
ejpam-5587	607	11	.	.	PUNCT
ejpam-5587	608	1	for	for	ADP
ejpam-5587	608	2	any	any	DET
ejpam-5587	608	3	x	x	SYM
ejpam-5587	608	4	∈	∈	PROPN
ejpam-5587	608	5	ker(t	ker(t	NOUN
ejpam-5587	608	6	−	−	PROPN
ejpam-5587	608	7	µ1i	µ1i	PROPN
ejpam-5587	608	8	)	)	PUNCT
ejpam-5587	608	9	and	and	CCONJ
ejpam-5587	608	10	y	y	PROPN
ejpam-5587	608	11	∈	∈	PROPN
ejpam-5587	608	12	ker(t	ker(t	NOUN
ejpam-5587	608	13	−	−	PROPN
ejpam-5587	608	14	µ2i	µ2i	NOUN
ejpam-5587	608	15	)	)	PUNCT
ejpam-5587	608	16	,	,	PUNCT
ejpam-5587	608	17	consider	consider	VERB
ejpam-5587	608	18	the	the	DET
ejpam-5587	608	19	set	set	NOUN
ejpam-5587	608	20	m	m	NOUN
ejpam-5587	608	21	=	=	PUNCT
ejpam-5587	608	22	span{x	span{x	PROPN
ejpam-5587	608	23	,	,	PUNCT
ejpam-5587	608	24	y	y	NOUN
ejpam-5587	608	25	}	}	PUNCT
ejpam-5587	608	26	.	.	PUNCT
ejpam-5587	609	1	as	as	SCONJ
ejpam-5587	609	2	m	m	PROPN
ejpam-5587	609	3	is	be	AUX
ejpam-5587	609	4	invariant	invariant	ADJ
ejpam-5587	609	5	subspace	subspace	NOUN
ejpam-5587	609	6	for	for	ADP
ejpam-5587	609	7	t	t	PROPN
ejpam-5587	609	8	,	,	PUNCT
ejpam-5587	609	9	it	it	PRON
ejpam-5587	609	10	follows	follow	VERB
ejpam-5587	609	11	that	that	SCONJ
ejpam-5587	609	12	t	t	NOUN
ejpam-5587	609	13	|m	|m	NOUN
ejpam-5587	609	14	is	be	AUX
ejpam-5587	609	15	totally	totally	ADV
ejpam-5587	609	16	paranormal	paranormal	ADJ
ejpam-5587	609	17	operator	operator	NOUN
ejpam-5587	609	18	and	and	CCONJ
ejpam-5587	609	19	∥t	∥t	PROPN
ejpam-5587	609	20	|m∥	|m∥	ADJ
ejpam-5587	609	21	=	=	PUNCT
ejpam-5587	609	22	|µ2|	|µ2|	NOUN
ejpam-5587	609	23	.	.	PUNCT
ejpam-5587	610	1	we	we	PRON
ejpam-5587	610	2	have	have	VERB
ejpam-5587	610	3	the	the	DET
ejpam-5587	610	4	following	following	NOUN
ejpam-5587	610	5	.	.	PUNCT
ejpam-5587	611	1	∥∥∥∥µn	∥∥∥∥µn	NUM
ejpam-5587	611	2	1	1	NUM
ejpam-5587	611	3	µn	µn	ADP
ejpam-5587	611	4	2	2	NUM
ejpam-5587	611	5	x+	x+	X
ejpam-5587	611	6	y	y	PROPN
ejpam-5587	611	7	∥∥∥∥	∥∥∥∥	NUM
ejpam-5587	612	1	=	=	SYM
ejpam-5587	612	2	1	1	NUM
ejpam-5587	612	3	|µn	|µn	NUM
ejpam-5587	612	4	2	2	NUM
ejpam-5587	612	5	|	|	ADV
ejpam-5587	612	6	∥µn	∥µn	NOUN
ejpam-5587	612	7	1x+	1x+	NUM
ejpam-5587	612	8	µn	µn	NOUN
ejpam-5587	612	9	2y∥	2y∥	NOUN
ejpam-5587	612	10	≤	≤	ADV
ejpam-5587	612	11	∥t	∥t	ADJ
ejpam-5587	612	12	|m∥n	|m∥n	X
ejpam-5587	612	13	|µn	|µn	X
ejpam-5587	612	14	2	2	NUM
ejpam-5587	612	15	|	|	NOUN
ejpam-5587	612	16	∥x+	∥x+	VERB
ejpam-5587	612	17	y∥	y∥	NOUN
ejpam-5587	612	18	=	=	PUNCT
ejpam-5587	612	19	∥x+	∥x+	PROPN
ejpam-5587	612	20	y∥	y∥	NOUN
ejpam-5587	612	21	taking	take	VERB
ejpam-5587	612	22	the	the	DET
ejpam-5587	612	23	limit	limit	NOUN
ejpam-5587	612	24	n	n	X
ejpam-5587	612	25	→	→	SYM
ejpam-5587	612	26	+	+	NOUN
ejpam-5587	612	27	∞	∞	PROPN
ejpam-5587	612	28	,	,	PUNCT
ejpam-5587	612	29	we	we	PRON
ejpam-5587	612	30	get	get	VERB
ejpam-5587	612	31	∥y∥	∥y∥	NOUN
ejpam-5587	612	32	≤	≤	NOUN
ejpam-5587	612	33	∥x+	∥x+	X
ejpam-5587	612	34	y∥	y∥	NOUN
ejpam-5587	612	35	,	,	PUNCT
ejpam-5587	612	36	for	for	ADP
ejpam-5587	612	37	every	every	DET
ejpam-5587	612	38	x	x	SYM
ejpam-5587	612	39	∈	∈	PROPN
ejpam-5587	612	40	ker(t	ker(t	NOUN
ejpam-5587	612	41	−	−	PROPN
ejpam-5587	612	42	µ1i	µ1i	PROPN
ejpam-5587	612	43	)	)	PUNCT
ejpam-5587	612	44	and	and	CCONJ
ejpam-5587	612	45	y	y	PROPN
ejpam-5587	612	46	∈	∈	PROPN
ejpam-5587	612	47	ker(t	ker(t	NOUN
ejpam-5587	612	48	−	−	PROPN
ejpam-5587	612	49	µ2i	µ2i	NOUN
ejpam-5587	612	50	)	)	PUNCT
ejpam-5587	612	51	.	.	PUNCT
ejpam-5587	613	1	hence	hence	ADV
ejpam-5587	613	2	ker(t	ker(t	NOUN
ejpam-5587	613	3	−	−	PROPN
ejpam-5587	613	4	µ2i	µ2i	ADJ
ejpam-5587	613	5	)	)	PUNCT
ejpam-5587	613	6	is	be	AUX
ejpam-5587	613	7	orthogonal	orthogonal	ADJ
ejpam-5587	613	8	to	to	ADP
ejpam-5587	613	9	ker(t	ker(t	PROPN
ejpam-5587	613	10	−	−	PROPN
ejpam-5587	613	11	µ1i	µ1i	PROPN
ejpam-5587	613	12	)	)	PUNCT
ejpam-5587	613	13	.	.	PUNCT
ejpam-5587	614	1	next	next	ADV
ejpam-5587	614	2	,	,	PUNCT
ejpam-5587	614	3	if	if	SCONJ
ejpam-5587	614	4	|µ1|	|µ1|	NOUN
ejpam-5587	614	5	=	=	SYM
ejpam-5587	614	6	|µ2|	|µ2|	NOUN
ejpam-5587	614	7	,	,	PUNCT
ejpam-5587	614	8	then	then	ADV
ejpam-5587	614	9	for	for	ADP
ejpam-5587	614	10	every	every	DET
ejpam-5587	614	11	n	n	PRON
ejpam-5587	614	12	∈	∈	NOUN
ejpam-5587	614	13	n∥∥∥∥(µ1	n∥∥∥∥(µ1	NOUN
ejpam-5587	614	14	+	+	CCONJ
ejpam-5587	614	15	µ2	µ2	PROPN
ejpam-5587	614	16	2µ2	2µ2	NUM
ejpam-5587	614	17	)	)	PUNCT
ejpam-5587	614	18	n	n	X
ejpam-5587	614	19	x+	x+	VERB
ejpam-5587	614	20	y	y	PROPN
ejpam-5587	614	21	∥∥∥∥	∥∥∥∥	NUM
ejpam-5587	614	22	=	=	SYM
ejpam-5587	615	1	∥∥∥∥(µ1	∥∥∥∥(µ1	PROPN
ejpam-5587	615	2	+	+	CCONJ
ejpam-5587	615	3	µ2	µ2	ADJ
ejpam-5587	615	4	)	)	PUNCT
ejpam-5587	615	5	nx+	nx+	NOUN
ejpam-5587	615	6	(	(	PUNCT
ejpam-5587	615	7	µ1	µ1	PROPN
ejpam-5587	615	8	+	+	X
ejpam-5587	615	9	µ2	µ2	PROPN
ejpam-5587	615	10	)	)	PUNCT
ejpam-5587	615	11	ny	ny	NOUN
ejpam-5587	616	1	(	(	PUNCT
ejpam-5587	616	2	2µ2)n	2µ2)n	NUM
ejpam-5587	616	3	∥∥∥∥	∥∥∥∥	SYM
ejpam-5587	616	4	≤	≤	NUM
ejpam-5587	616	5	1	1	NUM
ejpam-5587	616	6	(	(	PUNCT
ejpam-5587	616	7	2|µ2|)n	2|µ2|)n	PROPN
ejpam-5587	616	8	n∑	n∑	NOUN
ejpam-5587	616	9	j=0	j=0	PROPN
ejpam-5587	616	10	(	(	PUNCT
ejpam-5587	616	11	n	n	X
ejpam-5587	616	12	j	j	PROPN
ejpam-5587	616	13	)	)	PUNCT
ejpam-5587	617	1	|µ2|j	|µ2|j	PROPN
ejpam-5587	617	2	∥∥∥µn−j	∥∥∥µn−j	ADJ
ejpam-5587	617	3	1	1	NUM
ejpam-5587	617	4	x+	x+	PUNCT
ejpam-5587	617	5	µn−j	µn−j	NOUN
ejpam-5587	617	6	2	2	NUM
ejpam-5587	617	7	y	y	PROPN
ejpam-5587	617	8	∥∥∥	∥∥∥	PROPN
ejpam-5587	617	9	=	=	SYM
ejpam-5587	617	10	1	1	NUM
ejpam-5587	617	11	(	(	PUNCT
ejpam-5587	617	12	2|µ2|)n	2|µ2|)n	PROPN
ejpam-5587	617	13	n∑	n∑	NOUN
ejpam-5587	617	14	j=0	j=0	PROPN
ejpam-5587	617	15	(	(	PUNCT
ejpam-5587	617	16	n	n	X
ejpam-5587	617	17	j	j	PROPN
ejpam-5587	617	18	)	)	PUNCT
ejpam-5587	617	19	|µ2|j	|µ2|j	PROPN
ejpam-5587	617	20	∥∥∥(t	∥∥∥(t	NUM
ejpam-5587	617	21	|m	|m	NOUN
ejpam-5587	617	22	)	)	PUNCT
ejpam-5587	617	23	n−j	n−j	X
ejpam-5587	617	24	(	(	PUNCT
ejpam-5587	617	25	x+	x+	X
ejpam-5587	617	26	y	y	PROPN
ejpam-5587	617	27	)	)	PUNCT
ejpam-5587	617	28	∥∥∥	∥∥∥	PROPN
ejpam-5587	617	29	≤	≤	NUM
ejpam-5587	617	30	1	1	NUM
ejpam-5587	617	31	(	(	PUNCT
ejpam-5587	617	32	2|µ2|)n	2|µ2|)n	PROPN
ejpam-5587	617	33	n∑	n∑	NOUN
ejpam-5587	617	34	j=0	j=0	PROPN
ejpam-5587	617	35	(	(	PUNCT
ejpam-5587	617	36	n	n	X
ejpam-5587	617	37	j	j	PROPN
ejpam-5587	617	38	)	)	PUNCT
ejpam-5587	618	1	|µ2|n	|µ2|n	PROPN
ejpam-5587	618	2	∥x+	∥x+	VERB
ejpam-5587	618	3	y∥	y∥	NOUN
ejpam-5587	618	4	=	=	PUNCT
ejpam-5587	618	5	∥x+	∥x+	X
ejpam-5587	618	6	y∥	y∥	NOUN
ejpam-5587	618	7	.	.	PUNCT
ejpam-5587	619	1	as	as	ADP
ejpam-5587	619	2	µ1	µ1	PROPN
ejpam-5587	619	3	̸=	̸=	PROPN
ejpam-5587	619	4	µ2	µ2	NOUN
ejpam-5587	619	5	,	,	PUNCT
ejpam-5587	619	6	we	we	PRON
ejpam-5587	619	7	have	have	VERB
ejpam-5587	619	8	∣∣∣µ1+µ2	∣∣∣µ1+µ2	PROPN
ejpam-5587	619	9	2µ2	2µ2	NUM
ejpam-5587	619	10	∣∣∣	∣∣∣	NOUN
ejpam-5587	619	11	<	<	X
ejpam-5587	619	12	1	1	NUM
ejpam-5587	619	13	.	.	PUNCT
ejpam-5587	620	1	now	now	ADV
ejpam-5587	620	2	as	as	ADP
ejpam-5587	620	3	n	n	X
ejpam-5587	620	4	→	→	SYM
ejpam-5587	620	5	+	+	NUM
ejpam-5587	620	6	∞	∞	PROPN
ejpam-5587	620	7	in	in	ADP
ejpam-5587	620	8	the	the	DET
ejpam-5587	620	9	above	above	ADJ
ejpam-5587	620	10	inequality	inequality	NOUN
ejpam-5587	620	11	we	we	PRON
ejpam-5587	620	12	get	get	VERB
ejpam-5587	620	13	that	that	SCONJ
ejpam-5587	620	14	∥y∥	∥y∥	NOUN
ejpam-5587	620	15	≤	≤	NOUN
ejpam-5587	620	16	∥x+	∥x+	PUNCT
ejpam-5587	621	1	y∥.	y∥.	NOUN
ejpam-5587	621	2	this	this	PRON
ejpam-5587	621	3	proves	prove	VERB
ejpam-5587	621	4	the	the	DET
ejpam-5587	621	5	result	result	NOUN
ejpam-5587	621	6	s.	s.	PROPN
ejpam-5587	621	7	alnabulsi	alnabulsi	PROPN
ejpam-5587	621	8	,	,	PUNCT
ejpam-5587	621	9	m.h.m	m.h.m	PROPN
ejpam-5587	621	10	.	.	PUNCT
ejpam-5587	621	11	rashid	rashid	PROPN
ejpam-5587	621	12	/	/	SYM
ejpam-5587	621	13	eur	eur	PROPN
ejpam-5587	621	14	.	.	PUNCT
ejpam-5587	622	1	j.	j.	PROPN
ejpam-5587	622	2	pure	pure	PROPN
ejpam-5587	622	3	appl	appl	PROPN
ejpam-5587	622	4	.	.	PROPN
ejpam-5587	622	5	math	math	PROPN
ejpam-5587	622	6	,	,	PUNCT
ejpam-5587	622	7	18	18	NUM
ejpam-5587	622	8	(	(	PUNCT
ejpam-5587	622	9	1	1	NUM
ejpam-5587	622	10	)	)	PUNCT
ejpam-5587	622	11	(	(	PUNCT
ejpam-5587	622	12	2025	2025	NUM
ejpam-5587	622	13	)	)	PUNCT
ejpam-5587	622	14	,	,	PUNCT
ejpam-5587	622	15	5587	5587	NUM
ejpam-5587	622	16	17	17	NUM
ejpam-5587	622	17	of	of	ADP
ejpam-5587	622	18	20	20	NUM
ejpam-5587	622	19	proposition	proposition	NOUN
ejpam-5587	622	20	7	7	NUM
ejpam-5587	622	21	.	.	PUNCT
ejpam-5587	623	1	let	let	VERB
ejpam-5587	623	2	t	t	PROPN
ejpam-5587	623	3	∈	∈	PROPN
ejpam-5587	623	4	l(h	l(h	PROPN
ejpam-5587	623	5	)	)	PUNCT
ejpam-5587	623	6	be	be	AUX
ejpam-5587	623	7	a	a	DET
ejpam-5587	623	8	densely	densely	ADV
ejpam-5587	623	9	defined	define	VERB
ejpam-5587	623	10	closed	close	VERB
ejpam-5587	623	11	totally	totally	ADV
ejpam-5587	623	12	paranormal	paranormal	ADJ
ejpam-5587	623	13	operator	operator	NOUN
ejpam-5587	623	14	and	and	CCONJ
ejpam-5587	623	15	t	t	PROPN
ejpam-5587	623	16	2	2	NUM
ejpam-5587	623	17	be	be	AUX
ejpam-5587	623	18	a	a	DET
ejpam-5587	623	19	compact	compact	ADJ
ejpam-5587	623	20	operator	operator	NOUN
ejpam-5587	623	21	.	.	PUNCT
ejpam-5587	624	1	then	then	ADV
ejpam-5587	624	2	t	t	PROPN
ejpam-5587	624	3	is	be	AUX
ejpam-5587	624	4	also	also	ADV
ejpam-5587	624	5	compact	compact	ADJ
ejpam-5587	624	6	and	and	CCONJ
ejpam-5587	624	7	normal	normal	ADJ
ejpam-5587	624	8	.	.	PUNCT
ejpam-5587	625	1	proof	proof	NOUN
ejpam-5587	625	2	.	.	PUNCT
ejpam-5587	626	1	assume	assume	VERB
ejpam-5587	626	2	that	that	SCONJ
ejpam-5587	626	3	t	t	PROPN
ejpam-5587	626	4	is	be	AUX
ejpam-5587	626	5	a	a	DET
ejpam-5587	626	6	totally	totally	ADV
ejpam-5587	626	7	paranormal	paranormal	ADJ
ejpam-5587	626	8	operator	operator	NOUN
ejpam-5587	626	9	.	.	PUNCT
ejpam-5587	627	1	hence	hence	ADV
ejpam-5587	627	2	,	,	PUNCT
ejpam-5587	627	3	∥tx∥2	∥tx∥2	PROPN
ejpam-5587	627	4	≤	≤	NUM
ejpam-5587	627	5	∥∥t	∥∥t	VERB
ejpam-5587	627	6	2x	2x	NUM
ejpam-5587	627	7	∥∥	∥∥	PROPN
ejpam-5587	627	8	∥x∥	∥x∥	NOUN
ejpam-5587	627	9	for	for	ADP
ejpam-5587	627	10	every	every	DET
ejpam-5587	627	11	x	x	PROPN
ejpam-5587	627	12	∈	∈	PROPN
ejpam-5587	627	13	d(t	d(t	PROPN
ejpam-5587	627	14	2	2	NUM
ejpam-5587	627	15	)	)	PUNCT
ejpam-5587	627	16	.	.	PUNCT
ejpam-5587	628	1	(	(	PUNCT
ejpam-5587	628	2	6	6	X
ejpam-5587	628	3	)	)	PUNCT
ejpam-5587	628	4	let	let	AUX
ejpam-5587	628	5	{	{	PUNCT
ejpam-5587	628	6	xm	xm	NOUN
ejpam-5587	628	7	}	}	PUNCT
ejpam-5587	628	8	∈	∈	NOUN
ejpam-5587	628	9	h	h	NOUN
ejpam-5587	628	10	be	be	VERB
ejpam-5587	628	11	weakly	weakly	ADV
ejpam-5587	628	12	convergent	convergent	ADJ
ejpam-5587	628	13	sequence	sequence	NOUN
ejpam-5587	628	14	with	with	ADP
ejpam-5587	628	15	limit	limit	NOUN
ejpam-5587	628	16	0	0	NUM
ejpam-5587	628	17	in	in	ADP
ejpam-5587	628	18	d(t	d(t	PROPN
ejpam-5587	628	19	)	)	PUNCT
ejpam-5587	628	20	.	.	PUNCT
ejpam-5587	629	1	from	from	ADP
ejpam-5587	629	2	the	the	DET
ejpam-5587	629	3	compactness	compactness	NOUN
ejpam-5587	629	4	of	of	ADP
ejpam-5587	629	5	t	t	PROPN
ejpam-5587	629	6	2	2	NUM
ejpam-5587	629	7	and	and	CCONJ
ejpam-5587	629	8	the	the	DET
ejpam-5587	629	9	relation	relation	NOUN
ejpam-5587	629	10	(	(	PUNCT
ejpam-5587	629	11	6	6	NUM
ejpam-5587	629	12	)	)	PUNCT
ejpam-5587	629	13	we	we	PRON
ejpam-5587	629	14	get	get	VERB
ejpam-5587	629	15	the	the	DET
ejpam-5587	629	16	following	follow	VERB
ejpam-5587	629	17	relation	relation	NOUN
ejpam-5587	629	18	:	:	PUNCT
ejpam-5587	629	19	∥txm∥2	∥txm∥2	PROPN
ejpam-5587	629	20	→	→	SYM
ejpam-5587	629	21	0	0	NUM
ejpam-5587	629	22	,	,	PUNCT
ejpam-5587	629	23	m	m	VERB
ejpam-5587	629	24	→	→	SYM
ejpam-5587	629	25	+	+	PROPN
ejpam-5587	629	26	∞.	∞.	PROPN
ejpam-5587	629	27	from	from	ADP
ejpam-5587	629	28	the	the	DET
ejpam-5587	629	29	last	last	ADJ
ejpam-5587	629	30	relation	relation	NOUN
ejpam-5587	629	31	it	it	PRON
ejpam-5587	629	32	follows	follow	VERB
ejpam-5587	629	33	that	that	SCONJ
ejpam-5587	629	34	t	t	PROPN
ejpam-5587	629	35	is	be	AUX
ejpam-5587	629	36	compact	compact	ADJ
ejpam-5587	629	37	.	.	PUNCT
ejpam-5587	630	1	since	since	SCONJ
ejpam-5587	630	2	t	t	PROPN
ejpam-5587	630	3	is	be	AUX
ejpam-5587	630	4	compact	compact	ADJ
ejpam-5587	630	5	σ(t	σ(t	NOUN
ejpam-5587	630	6	)	)	PUNCT
ejpam-5587	630	7	is	be	AUX
ejpam-5587	630	8	finite	finite	NOUN
ejpam-5587	630	9	set	set	VERB
ejpam-5587	630	10	or	or	CCONJ
ejpam-5587	630	11	countable	countable	VERB
ejpam-5587	630	12	infinite	infinite	NOUN
ejpam-5587	630	13	with	with	ADP
ejpam-5587	630	14	0	0	NUM
ejpam-5587	630	15	as	as	ADP
ejpam-5587	630	16	the	the	DET
ejpam-5587	630	17	unique	unique	ADJ
ejpam-5587	630	18	limit	limit	NOUN
ejpam-5587	630	19	point	point	NOUN
ejpam-5587	630	20	of	of	ADP
ejpam-5587	630	21	it	it	PRON
ejpam-5587	630	22	.	.	PUNCT
ejpam-5587	631	1	let	let	VERB
ejpam-5587	631	2	σ(t	σ(t	PROPN
ejpam-5587	631	3	)	)	PUNCT
ejpam-5587	631	4	\	\	PROPN
ejpam-5587	632	1	{	{	PUNCT
ejpam-5587	632	2	0	0	NUM
ejpam-5587	632	3	}	}	PUNCT
ejpam-5587	632	4	=	=	SYM
ejpam-5587	632	5	{	{	PUNCT
ejpam-5587	632	6	λn	λn	NOUN
ejpam-5587	632	7	}	}	PUNCT
ejpam-5587	632	8	with	with	ADP
ejpam-5587	632	9	|λ1|	|λ1|	ADP
ejpam-5587	632	10	≥	≥	PROPN
ejpam-5587	632	11	|λ2|	|λ2|	VERB
ejpam-5587	632	12	≥	≥	X
ejpam-5587	632	13	·	·	PUNCT
ejpam-5587	632	14	·	·	PUNCT
ejpam-5587	632	15	·	·	PUNCT
ejpam-5587	632	16	≥	≥	NUM
ejpam-5587	632	17	|λn|	|λn|	PROPN
ejpam-5587	632	18	≥	≥	PRON
ejpam-5587	632	19	|λn+1|	|λn+1|	PROPN
ejpam-5587	632	20	≥	≥	NUM
ejpam-5587	632	21	·	·	PUNCT
ejpam-5587	632	22	·	·	PUNCT
ejpam-5587	632	23	·	·	PUNCT
ejpam-5587	632	24	≥	≥	NOUN
ejpam-5587	632	25	0	0	NUM
ejpam-5587	632	26	,	,	PUNCT
ejpam-5587	632	27	and	and	CCONJ
ejpam-5587	632	28	λn	λn	X
ejpam-5587	632	29	→	→	SYM
ejpam-5587	632	30	0	0	NUM
ejpam-5587	632	31	(	(	PUNCT
ejpam-5587	632	32	n	n	NOUN
ejpam-5587	632	33	→	→	SYM
ejpam-5587	632	34	+	+	NOUN
ejpam-5587	632	35	∞	∞	NUM
ejpam-5587	632	36	)	)	PUNCT
ejpam-5587	632	37	.	.	PUNCT
ejpam-5587	633	1	by	by	ADP
ejpam-5587	633	2	the	the	DET
ejpam-5587	633	3	compactness	compactness	NOUN
ejpam-5587	633	4	of	of	ADP
ejpam-5587	633	5	t	t	PROPN
ejpam-5587	633	6	or	or	CCONJ
ejpam-5587	633	7	isoloidness	isoloidness	NOUN
ejpam-5587	633	8	of	of	ADP
ejpam-5587	633	9	t	t	PROPN
ejpam-5587	633	10	,	,	PUNCT
ejpam-5587	633	11	λn	λn	PROPN
ejpam-5587	633	12	∈	∈	PROPN
ejpam-5587	633	13	σp(t	σp(t	PUNCT
ejpam-5587	633	14	)	)	PUNCT
ejpam-5587	633	15	and	and	CCONJ
ejpam-5587	633	16	dim	dim	ADJ
ejpam-5587	633	17	ker(t	ker(t	NOUN
ejpam-5587	633	18	−	−	PROPN
ejpam-5587	633	19	λn	λn	NOUN
ejpam-5587	633	20	)	)	PUNCT
ejpam-5587	633	21	<	<	X
ejpam-5587	634	1	+	+	ADJ
ejpam-5587	634	2	∞	∞	NUM
ejpam-5587	634	3	for	for	ADP
ejpam-5587	634	4	all	all	DET
ejpam-5587	634	5	n.	n.	NOUN
ejpam-5587	634	6	since	since	SCONJ
ejpam-5587	634	7	ker(t	ker(t	PROPN
ejpam-5587	634	8	−	−	PROPN
ejpam-5587	634	9	λn	λn	NOUN
ejpam-5587	634	10	)	)	PUNCT
ejpam-5587	634	11	⊂	⊂	PROPN
ejpam-5587	634	12	ker(t	ker(t	VERB
ejpam-5587	634	13	−	−	PROPN
ejpam-5587	634	14	λn	λn	NOUN
ejpam-5587	634	15	)	)	PUNCT
ejpam-5587	634	16	∗	∗	NOUN
ejpam-5587	634	17	,	,	PUNCT
ejpam-5587	634	18	m	m	VERB
ejpam-5587	634	19	:	:	PUNCT
ejpam-5587	634	20	=	=	SYM
ejpam-5587	634	21	⊕+∞	⊕+∞	PROPN
ejpam-5587	634	22	n=1	n=1	ADP
ejpam-5587	634	23	ker(t	ker(t	PROPN
ejpam-5587	634	24	−	−	PROPN
ejpam-5587	634	25	λn	λn	NOUN
ejpam-5587	634	26	)	)	PUNCT
ejpam-5587	634	27	reduces	reduce	VERB
ejpam-5587	634	28	t	t	NOUN
ejpam-5587	634	29	,	,	PUNCT
ejpam-5587	634	30	and	and	CCONJ
ejpam-5587	634	31	t	t	PROPN
ejpam-5587	634	32	is	be	AUX
ejpam-5587	634	33	of	of	ADP
ejpam-5587	634	34	the	the	DET
ejpam-5587	634	35	form	form	NOUN
ejpam-5587	634	36	t	t	NOUN
ejpam-5587	634	37	=	=	SYM
ejpam-5587	634	38	(	(	PUNCT
ejpam-5587	634	39	+	+	PROPN
ejpam-5587	634	40	∞⊕	∞⊕	PROPN
ejpam-5587	634	41	n=1	n=1	PROPN
ejpam-5587	634	42	λn	λn	PROPN
ejpam-5587	634	43	)	)	PUNCT
ejpam-5587	634	44	⊕	⊕	PROPN
ejpam-5587	634	45	t	t	NOUN
ejpam-5587	634	46	′	′	NOUN
ejpam-5587	634	47	on	on	ADP
ejpam-5587	634	48	h	h	NOUN
ejpam-5587	635	1	=	=	PUNCT
ejpam-5587	635	2	m⊕m⊥.	m⊕m⊥.	PUNCT
ejpam-5587	635	3	by	by	ADP
ejpam-5587	635	4	the	the	DET
ejpam-5587	635	5	construction	construction	NOUN
ejpam-5587	635	6	,	,	PUNCT
ejpam-5587	635	7	t	t	PROPN
ejpam-5587	635	8	′	′	NUM
ejpam-5587	635	9	is	be	AUX
ejpam-5587	635	10	totally	totally	ADV
ejpam-5587	635	11	paranormal	paranormal	ADJ
ejpam-5587	635	12	and	and	CCONJ
ejpam-5587	635	13	σ(t	σ(t	PROPN
ejpam-5587	635	14	′	′	NUM
ejpam-5587	635	15	)	)	PUNCT
ejpam-5587	635	16	=	=	PRON
ejpam-5587	636	1	{	{	PUNCT
ejpam-5587	636	2	0	0	NUM
ejpam-5587	636	3	}	}	PUNCT
ejpam-5587	636	4	hence	hence	ADV
ejpam-5587	636	5	t	t	NOUN
ejpam-5587	636	6	′	′	NUM
ejpam-5587	637	1	=	=	NOUN
ejpam-5587	637	2	0	0	X
ejpam-5587	637	3	.	.	PUNCT
ejpam-5587	638	1	this	this	PRON
ejpam-5587	638	2	shows	show	VERB
ejpam-5587	638	3	that	that	SCONJ
ejpam-5587	638	4	t	t	NOUN
ejpam-5587	638	5	=	=	PUNCT
ejpam-5587	638	6	(	(	PUNCT
ejpam-5587	638	7	+	+	PROPN
ejpam-5587	638	8	∞⊕	∞⊕	PROPN
ejpam-5587	638	9	n=1	n=1	PROPN
ejpam-5587	638	10	λn	λn	PROPN
ejpam-5587	638	11	)	)	PUNCT
ejpam-5587	638	12	⊕	⊕	PROPN
ejpam-5587	638	13	0	0	PUNCT
ejpam-5587	639	1	and	and	CCONJ
ejpam-5587	639	2	it	it	PRON
ejpam-5587	639	3	is	be	AUX
ejpam-5587	639	4	normal	normal	ADJ
ejpam-5587	639	5	.	.	PUNCT
ejpam-5587	640	1	theorem	theorem	VERB
ejpam-5587	640	2	14	14	NUM
ejpam-5587	640	3	.	.	PUNCT
ejpam-5587	641	1	let	let	VERB
ejpam-5587	641	2	t	t	PROPN
ejpam-5587	641	3	∈	∈	PROPN
ejpam-5587	641	4	l(h	l(h	PROPN
ejpam-5587	641	5	)	)	PUNCT
ejpam-5587	641	6	be	be	AUX
ejpam-5587	641	7	a	a	DET
ejpam-5587	641	8	densely	densely	ADV
ejpam-5587	641	9	defined	define	VERB
ejpam-5587	641	10	closed	close	VERB
ejpam-5587	641	11	totally	totally	ADV
ejpam-5587	641	12	paranormal	paranormal	ADJ
ejpam-5587	641	13	operator	operator	NOUN
ejpam-5587	641	14	with	with	ADP
ejpam-5587	641	15	σw(t	σw(t	PUNCT
ejpam-5587	641	16	)	)	PUNCT
ejpam-5587	641	17	=	=	PUNCT
ejpam-5587	641	18	{	{	PUNCT
ejpam-5587	641	19	0	0	NUM
ejpam-5587	641	20	}	}	PUNCT
ejpam-5587	641	21	.	.	PUNCT
ejpam-5587	642	1	then	then	ADV
ejpam-5587	642	2	t	t	PROPN
ejpam-5587	642	3	is	be	AUX
ejpam-5587	642	4	a	a	DET
ejpam-5587	642	5	compact	compact	ADJ
ejpam-5587	642	6	normal	normal	ADJ
ejpam-5587	642	7	operator	operator	NOUN
ejpam-5587	642	8	.	.	PUNCT
ejpam-5587	643	1	proof	proof	NOUN
ejpam-5587	643	2	.	.	PUNCT
ejpam-5587	644	1	by	by	ADP
ejpam-5587	644	2	theorem	theorem	NOUN
ejpam-5587	644	3	12	12	NUM
ejpam-5587	644	4	,	,	PUNCT
ejpam-5587	644	5	t	t	PROPN
ejpam-5587	644	6	satisfy	satisfy	PROPN
ejpam-5587	644	7	weyl	weyl	PROPN
ejpam-5587	644	8	’s	’s	PART
ejpam-5587	644	9	theorem	theorem	NOUN
ejpam-5587	644	10	and	and	CCONJ
ejpam-5587	644	11	this	this	PRON
ejpam-5587	644	12	implies	imply	VERB
ejpam-5587	644	13	that	that	SCONJ
ejpam-5587	644	14	each	each	DET
ejpam-5587	644	15	element	element	NOUN
ejpam-5587	644	16	in	in	ADP
ejpam-5587	644	17	σ(t	σ(t	PROPN
ejpam-5587	644	18	)	)	PUNCT
ejpam-5587	644	19	\	\	PROPN
ejpam-5587	644	20	σw(t	σw(t	PUNCT
ejpam-5587	644	21	)	)	PUNCT
ejpam-5587	645	1	=	=	SYM
ejpam-5587	645	2	σ(t	σ(t	PROPN
ejpam-5587	645	3	)	)	PUNCT
ejpam-5587	645	4	\	\	PROPN
ejpam-5587	645	5	{	{	PUNCT
ejpam-5587	645	6	0	0	NUM
ejpam-5587	645	7	}	}	PUNCT
ejpam-5587	645	8	is	be	AUX
ejpam-5587	645	9	an	an	DET
ejpam-5587	645	10	eigenvalue	eigenvalue	NOUN
ejpam-5587	645	11	of	of	ADP
ejpam-5587	645	12	t	t	PROPN
ejpam-5587	645	13	with	with	ADP
ejpam-5587	645	14	finite	finite	ADJ
ejpam-5587	645	15	multiplicity	multiplicity	NOUN
ejpam-5587	645	16	,	,	PUNCT
ejpam-5587	645	17	and	and	CCONJ
ejpam-5587	645	18	is	be	AUX
ejpam-5587	645	19	isolated	isolate	VERB
ejpam-5587	645	20	in	in	ADP
ejpam-5587	645	21	σ(t	σ(t	PROPN
ejpam-5587	645	22	)	)	PUNCT
ejpam-5587	645	23	.	.	PUNCT
ejpam-5587	646	1	hence	hence	ADV
ejpam-5587	646	2	σ(t	σ(t	PROPN
ejpam-5587	646	3	)	)	PUNCT
ejpam-5587	646	4	\	\	PROPN
ejpam-5587	646	5	{	{	PUNCT
ejpam-5587	646	6	0	0	NUM
ejpam-5587	646	7	}	}	PUNCT
ejpam-5587	646	8	is	be	AUX
ejpam-5587	646	9	a	a	DET
ejpam-5587	646	10	finite	finite	NOUN
ejpam-5587	646	11	set	set	NOUN
ejpam-5587	646	12	or	or	CCONJ
ejpam-5587	646	13	a	a	DET
ejpam-5587	646	14	countable	countable	ADJ
ejpam-5587	646	15	set	set	NOUN
ejpam-5587	646	16	with	with	ADP
ejpam-5587	646	17	0	0	NUM
ejpam-5587	646	18	as	as	ADP
ejpam-5587	646	19	its	its	PRON
ejpam-5587	646	20	only	only	ADJ
ejpam-5587	646	21	accumulation	accumulation	NOUN
ejpam-5587	646	22	point	point	NOUN
ejpam-5587	646	23	.	.	PUNCT
ejpam-5587	647	1	put	put	VERB
ejpam-5587	647	2	σ(t	σ(t	PROPN
ejpam-5587	647	3	)	)	PUNCT
ejpam-5587	647	4	\	\	PROPN
ejpam-5587	647	5	{	{	PUNCT
ejpam-5587	647	6	λn	λn	NOUN
ejpam-5587	647	7	}	}	PUNCT
ejpam-5587	647	8	,	,	PUNCT
ejpam-5587	648	1	where	where	SCONJ
ejpam-5587	648	2	λn	λn	PROPN
ejpam-5587	648	3	̸=	̸=	PROPN
ejpam-5587	648	4	λm	λm	ADP
ejpam-5587	648	5	whenever	whenever	SCONJ
ejpam-5587	648	6	n	n	PRON
ejpam-5587	648	7	̸=	̸=	PROPN
ejpam-5587	648	8	m	m	PROPN
ejpam-5587	648	9	and	and	CCONJ
ejpam-5587	648	10	{	{	PUNCT
ejpam-5587	648	11	|λn|	|λn|	NOUN
ejpam-5587	648	12	}	}	PUNCT
ejpam-5587	648	13	is	be	AUX
ejpam-5587	648	14	a	a	DET
ejpam-5587	648	15	non	non	ADJ
ejpam-5587	648	16	-	-	ADJ
ejpam-5587	648	17	increasing	increasing	ADJ
ejpam-5587	648	18	sequence	sequence	NOUN
ejpam-5587	648	19	.	.	PUNCT
ejpam-5587	649	1	since	since	SCONJ
ejpam-5587	649	2	t	t	PROPN
ejpam-5587	649	3	is	be	AUX
ejpam-5587	649	4	normaloid	normaloid	NOUN
ejpam-5587	649	5	,	,	PUNCT
ejpam-5587	649	6	we	we	PRON
ejpam-5587	649	7	have	have	VERB
ejpam-5587	649	8	|λ1|	|λ1|	NOUN
ejpam-5587	649	9	=	=	SYM
ejpam-5587	649	10	∥t∥	∥t∥	NOUN
ejpam-5587	649	11	.	.	PUNCT
ejpam-5587	650	1	by	by	ADP
ejpam-5587	650	2	corollary	corollary	ADJ
ejpam-5587	650	3	3	3	NUM
ejpam-5587	650	4	,	,	PUNCT
ejpam-5587	650	5	we	we	PRON
ejpam-5587	650	6	have	have	VERB
ejpam-5587	650	7	(	(	PUNCT
ejpam-5587	650	8	t−λ1i)x	t−λ1i)x	NUM
ejpam-5587	650	9	=	=	SYM
ejpam-5587	650	10	0	0	NUM
ejpam-5587	650	11	implies	imply	VERB
ejpam-5587	650	12	(	(	PUNCT
ejpam-5587	650	13	t	t	PROPN
ejpam-5587	650	14	−	−	PROPN
ejpam-5587	650	15	λ1i	λ1i	NOUN
ejpam-5587	650	16	)	)	PUNCT
ejpam-5587	651	1	∗x	∗x	PROPN
ejpam-5587	651	2	=	=	SYM
ejpam-5587	651	3	0	0	X
ejpam-5587	651	4	.	.	PUNCT
ejpam-5587	652	1	hence	hence	ADV
ejpam-5587	652	2	ker(t	ker(t	NOUN
ejpam-5587	652	3	−	−	PROPN
ejpam-5587	652	4	λ1i	λ1i	NOUN
ejpam-5587	652	5	)	)	PUNCT
ejpam-5587	652	6	is	be	AUX
ejpam-5587	652	7	a	a	DET
ejpam-5587	652	8	reducing	reduce	VERB
ejpam-5587	652	9	subspace	subspace	NOUN
ejpam-5587	652	10	of	of	ADP
ejpam-5587	652	11	t	t	PROPN
ejpam-5587	652	12	.	.	PUNCT
ejpam-5587	653	1	let	let	VERB
ejpam-5587	653	2	e1	e1	NOUN
ejpam-5587	653	3	be	be	AUX
ejpam-5587	653	4	the	the	DET
ejpam-5587	653	5	orthogonal	orthogonal	ADJ
ejpam-5587	653	6	projection	projection	NOUN
ejpam-5587	653	7	onto	onto	ADP
ejpam-5587	653	8	ker(t	ker(t	PROPN
ejpam-5587	653	9	−	−	PROPN
ejpam-5587	653	10	λ1i	λ1i	NOUN
ejpam-5587	653	11	)	)	PUNCT
ejpam-5587	653	12	.	.	PUNCT
ejpam-5587	654	1	then	then	ADV
ejpam-5587	654	2	t	t	PROPN
ejpam-5587	654	3	=	=	PUNCT
ejpam-5587	655	1	λ1i	λ1i	PROPN
ejpam-5587	655	2	⊕	⊕	PROPN
ejpam-5587	655	3	t1	t1	VERB
ejpam-5587	655	4	on	on	ADP
ejpam-5587	655	5	h	h	NOUN
ejpam-5587	655	6	=	=	PUNCT
ejpam-5587	655	7	ran(e1	ran(e1	PROPN
ejpam-5587	655	8	)	)	PUNCT
ejpam-5587	655	9	⊕	⊕	PROPN
ejpam-5587	655	10	ran(i	ran(i	PROPN
ejpam-5587	655	11	−e1	−e1	PROPN
ejpam-5587	655	12	)	)	PUNCT
ejpam-5587	655	13	.	.	PUNCT
ejpam-5587	656	1	since	since	SCONJ
ejpam-5587	656	2	t1	t1	NOUN
ejpam-5587	656	3	is	be	AUX
ejpam-5587	656	4	totally	totally	ADV
ejpam-5587	656	5	paranormal	paranormal	ADJ
ejpam-5587	656	6	by	by	ADP
ejpam-5587	656	7	theorem	theorem	NOUN
ejpam-5587	656	8	5	5	NUM
ejpam-5587	656	9	(	(	PUNCT
ejpam-5587	656	10	i	i	NOUN
ejpam-5587	656	11	)	)	PUNCT
ejpam-5587	656	12	and	and	CCONJ
ejpam-5587	656	13	σp(t	σp(t	PUNCT
ejpam-5587	656	14	)	)	PUNCT
ejpam-5587	657	1	=	=	PUNCT
ejpam-5587	657	2	σp(t1)∪	σp(t1)∪	ADJ
ejpam-5587	657	3	{	{	PUNCT
ejpam-5587	657	4	λ1	λ1	PROPN
ejpam-5587	657	5	}	}	PUNCT
ejpam-5587	657	6	,	,	PUNCT
ejpam-5587	657	7	we	we	PRON
ejpam-5587	657	8	have	have	VERB
ejpam-5587	657	9	λ2	λ2	NOUN
ejpam-5587	657	10	∈	∈	PROPN
ejpam-5587	657	11	σp(t1	σp(t1	NOUN
ejpam-5587	657	12	)	)	PUNCT
ejpam-5587	657	13	.	.	PUNCT
ejpam-5587	658	1	by	by	ADP
ejpam-5587	658	2	the	the	DET
ejpam-5587	658	3	same	same	ADJ
ejpam-5587	658	4	argument	argument	NOUN
ejpam-5587	658	5	as	as	ADP
ejpam-5587	658	6	above	above	ADV
ejpam-5587	658	7	,	,	PUNCT
ejpam-5587	658	8	ker(t	ker(t	NOUN
ejpam-5587	658	9	−	−	PROPN
ejpam-5587	658	10	λ2i	λ2i	NOUN
ejpam-5587	658	11	)	)	PUNCT
ejpam-5587	658	12	=	=	SYM
ejpam-5587	659	1	ker(t1	ker(t1	PROPN
ejpam-5587	659	2	−	−	PROPN
ejpam-5587	659	3	λ2i	λ2i	PROPN
ejpam-5587	659	4	)	)	PUNCT
ejpam-5587	659	5	is	be	AUX
ejpam-5587	659	6	a	a	DET
ejpam-5587	659	7	finite	finite	ADJ
ejpam-5587	659	8	dimensional	dimensional	ADJ
ejpam-5587	659	9	reducing	reduce	VERB
ejpam-5587	659	10	subspace	subspace	NOUN
ejpam-5587	659	11	of	of	ADP
ejpam-5587	659	12	t	t	PROPN
ejpam-5587	659	13	which	which	PRON
ejpam-5587	659	14	is	be	AUX
ejpam-5587	659	15	included	include	VERB
ejpam-5587	659	16	in	in	ADP
ejpam-5587	659	17	ran(i	ran(i	PROPN
ejpam-5587	659	18	−	−	PROPN
ejpam-5587	659	19	e1	e1	PROPN
ejpam-5587	659	20	)	)	PUNCT
ejpam-5587	659	21	.	.	PUNCT
ejpam-5587	660	1	put	put	PROPN
ejpam-5587	660	2	e2	e2	PROPN
ejpam-5587	660	3	be	be	AUX
ejpam-5587	660	4	the	the	DET
ejpam-5587	660	5	othogonal	othogonal	ADJ
ejpam-5587	660	6	projection	projection	NOUN
ejpam-5587	660	7	onto	onto	ADP
ejpam-5587	660	8	ker(t	ker(t	PROPN
ejpam-5587	660	9	−	−	PROPN
ejpam-5587	660	10	λ2i	λ2i	NOUN
ejpam-5587	660	11	)	)	PUNCT
ejpam-5587	660	12	.	.	PUNCT
ejpam-5587	661	1	then	then	ADV
ejpam-5587	661	2	t	t	PROPN
ejpam-5587	661	3	=	=	SYM
ejpam-5587	661	4	λ1e1	λ1e1	PROPN
ejpam-5587	661	5	⊕	⊕	PROPN
ejpam-5587	661	6	λ2e2	λ2e2	PROPN
ejpam-5587	662	1	⊕	⊕	PROPN
ejpam-5587	662	2	t2	t2	PROPN
ejpam-5587	662	3	on	on	ADP
ejpam-5587	662	4	h	h	NOUN
ejpam-5587	662	5	=	=	SYM
ejpam-5587	662	6	ran(e1)⊕ran(e2)⊕ran(i−e1−e2	ran(e1)⊕ran(e2)⊕ran(i−e1−e2	NOUN
ejpam-5587	662	7	)	)	PUNCT
ejpam-5587	662	8	.	.	PUNCT
ejpam-5587	663	1	by	by	ADP
ejpam-5587	663	2	repeating	repeat	VERB
ejpam-5587	663	3	above	above	ADP
ejpam-5587	663	4	argument	argument	NOUN
ejpam-5587	663	5	,	,	PUNCT
ejpam-5587	663	6	each	each	DET
ejpam-5587	663	7	ker(t−λni	ker(t−λni	PROPN
ejpam-5587	663	8	)	)	PUNCT
ejpam-5587	663	9	s.	s.	PROPN
ejpam-5587	663	10	alnabulsi	alnabulsi	PROPN
ejpam-5587	663	11	,	,	PUNCT
ejpam-5587	663	12	m.h.m	m.h.m	PROPN
ejpam-5587	663	13	.	.	PUNCT
ejpam-5587	663	14	rashid	rashid	PROPN
ejpam-5587	663	15	/	/	SYM
ejpam-5587	663	16	eur	eur	PROPN
ejpam-5587	663	17	.	.	PUNCT
ejpam-5587	664	1	j.	j.	PROPN
ejpam-5587	664	2	pure	pure	PROPN
ejpam-5587	664	3	appl	appl	PROPN
ejpam-5587	664	4	.	.	PROPN
ejpam-5587	664	5	math	math	PROPN
ejpam-5587	664	6	,	,	PUNCT
ejpam-5587	664	7	18	18	NUM
ejpam-5587	664	8	(	(	PUNCT
ejpam-5587	664	9	1	1	NUM
ejpam-5587	664	10	)	)	PUNCT
ejpam-5587	664	11	(	(	PUNCT
ejpam-5587	664	12	2025	2025	NUM
ejpam-5587	664	13	)	)	PUNCT
ejpam-5587	664	14	,	,	PUNCT
ejpam-5587	664	15	5587	5587	NUM
ejpam-5587	664	16	18	18	NUM
ejpam-5587	664	17	of	of	ADP
ejpam-5587	664	18	20	20	NUM
ejpam-5587	664	19	is	be	AUX
ejpam-5587	664	20	a	a	DET
ejpam-5587	664	21	reducing	reduce	VERB
ejpam-5587	664	22	subspace	subspace	NOUN
ejpam-5587	664	23	of	of	ADP
ejpam-5587	664	24	t	t	PROPN
ejpam-5587	664	25	and	and	CCONJ
ejpam-5587	664	26	∥∥∥∥∥t	∥∥∥∥∥t	PROPN
ejpam-5587	665	1	−	−	PROPN
ejpam-5587	665	2	n⊕	n⊕	NOUN
ejpam-5587	665	3	k=1	k=1	X
ejpam-5587	665	4	λkek	λkek	PROPN
ejpam-5587	665	5	∥∥∥∥∥	∥∥∥∥∥	PROPN
ejpam-5587	665	6	=	=	SYM
ejpam-5587	665	7	∥tn∥	∥tn∥	NUM
ejpam-5587	666	1	=	=	SYM
ejpam-5587	666	2	|λn+1|	|λn+1|	PROPN
ejpam-5587	666	3	→	→	SYM
ejpam-5587	666	4	0	0	PROPN
ejpam-5587	666	5	as	as	ADP
ejpam-5587	666	6	n	n	PRON
ejpam-5587	666	7	→	→	PUNCT
ejpam-5587	666	8	+	+	PROPN
ejpam-5587	666	9	∞.	∞.	PROPN
ejpam-5587	666	10	here	here	ADV
ejpam-5587	666	11	ek	ek	PROPN
ejpam-5587	666	12	is	be	AUX
ejpam-5587	666	13	the	the	DET
ejpam-5587	666	14	orthogonal	orthogonal	ADJ
ejpam-5587	666	15	projection	projection	NOUN
ejpam-5587	666	16	onto	onto	ADP
ejpam-5587	666	17	ker(t	ker(t	PROPN
ejpam-5587	666	18	−	−	PROPN
ejpam-5587	666	19	λk	λk	NOUN
ejpam-5587	666	20	)	)	PUNCT
ejpam-5587	666	21	and	and	CCONJ
ejpam-5587	666	22	t	t	NOUN
ejpam-5587	666	23	=	=	SYM
ejpam-5587	666	24	(	(	PUNCT
ejpam-5587	666	25	n⊕	n⊕	NOUN
ejpam-5587	666	26	k=1	k=1	X
ejpam-5587	666	27	λkek	λkek	PROPN
ejpam-5587	666	28	)	)	PUNCT
ejpam-5587	666	29	⊕	⊕	PROPN
ejpam-5587	666	30	tn	tn	PROPN
ejpam-5587	666	31	on	on	ADP
ejpam-5587	666	32	h	h	NOUN
ejpam-5587	666	33	=	=	SYM
ejpam-5587	666	34	n⊕	n⊕	PROPN
ejpam-5587	667	1	k=1	k=1	X
ejpam-5587	667	2	ran(ek)⊕	ran(ek)⊕	PROPN
ejpam-5587	667	3	(	(	PUNCT
ejpam-5587	667	4	1−	1−	NUM
ejpam-5587	667	5	n∑	n∑	NOUN
ejpam-5587	667	6	k=1	k=1	PROPN
ejpam-5587	667	7	ran(ek	ran(ek	PROPN
ejpam-5587	667	8	)	)	PUNCT
ejpam-5587	667	9	.	.	PUNCT
ejpam-5587	668	1	hence	hence	ADV
ejpam-5587	668	2	t	t	NOUN
ejpam-5587	668	3	=	=	PUNCT
ejpam-5587	669	1	+	+	PROPN
ejpam-5587	669	2	∞⊕	∞⊕	X
ejpam-5587	669	3	k=1	k=1	X
ejpam-5587	669	4	λkek	λkek	PROPN
ejpam-5587	669	5	is	be	AUX
ejpam-5587	669	6	compact	compact	ADJ
ejpam-5587	669	7	and	and	CCONJ
ejpam-5587	669	8	normal	normal	ADJ
ejpam-5587	669	9	because	because	SCONJ
ejpam-5587	669	10	each	each	DET
ejpam-5587	669	11	ek	ek	NOUN
ejpam-5587	669	12	is	be	AUX
ejpam-5587	669	13	a	a	DET
ejpam-5587	669	14	finite	finite	ADJ
ejpam-5587	669	15	rank	rank	PROPN
ejpam-5587	669	16	orthogonal	orthogonal	PROPN
ejpam-5587	669	17	projection	projection	NOUN
ejpam-5587	669	18	which	which	PRON
ejpam-5587	669	19	satisfies	satisfy	VERB
ejpam-5587	669	20	eket	eket	NOUN
ejpam-5587	669	21	=	=	SYM
ejpam-5587	669	22	0	0	PUNCT
ejpam-5587	670	1	whenever	whenever	SCONJ
ejpam-5587	670	2	k	k	PROPN
ejpam-5587	670	3	̸=	̸=	PROPN
ejpam-5587	670	4	t	t	X
ejpam-5587	670	5	by	by	ADP
ejpam-5587	670	6	proposition	proposition	NOUN
ejpam-5587	670	7	6	6	NUM
ejpam-5587	670	8	and	and	CCONJ
ejpam-5587	670	9	λn	λn	X
ejpam-5587	670	10	→	→	SYM
ejpam-5587	670	11	0	0	PUNCT
ejpam-5587	670	12	as	as	ADP
ejpam-5587	670	13	n	n	PRON
ejpam-5587	670	14	→	→	PUNCT
ejpam-5587	670	15	+	+	PROPN
ejpam-5587	670	16	∞.	∞.	PROPN
ejpam-5587	670	17	5	5	NUM
ejpam-5587	670	18	.	.	PUNCT
ejpam-5587	671	1	conclusion	conclusion	NOUN
ejpam-5587	671	2	to	to	PART
ejpam-5587	671	3	conclude	conclude	VERB
ejpam-5587	672	1	,	,	PUNCT
ejpam-5587	672	2	this	this	DET
ejpam-5587	672	3	article	article	NOUN
ejpam-5587	672	4	provides	provide	VERB
ejpam-5587	672	5	a	a	DET
ejpam-5587	672	6	thorough	thorough	ADJ
ejpam-5587	672	7	analysis	analysis	NOUN
ejpam-5587	672	8	of	of	ADP
ejpam-5587	672	9	the	the	DET
ejpam-5587	672	10	spectral	spectral	ADJ
ejpam-5587	672	11	properties	property	NOUN
ejpam-5587	672	12	of	of	ADP
ejpam-5587	672	13	totally	totally	ADV
ejpam-5587	672	14	paranormal	paranormal	ADJ
ejpam-5587	672	15	closed	closed	ADJ
ejpam-5587	672	16	operators	operator	NOUN
ejpam-5587	672	17	within	within	ADP
ejpam-5587	672	18	hilbert	hilbert	PROPN
ejpam-5587	672	19	spaces	space	NOUN
ejpam-5587	672	20	.	.	PUNCT
ejpam-5587	673	1	the	the	DET
ejpam-5587	673	2	study	study	NOUN
ejpam-5587	673	3	goes	go	VERB
ejpam-5587	673	4	beyond	beyond	ADP
ejpam-5587	673	5	standard	standard	ADJ
ejpam-5587	673	6	constraints	constraint	NOUN
ejpam-5587	673	7	on	on	ADP
ejpam-5587	673	8	boundedness	boundedness	NOUN
ejpam-5587	673	9	,	,	PUNCT
ejpam-5587	673	10	also	also	ADV
ejpam-5587	673	11	considering	consider	VERB
ejpam-5587	673	12	closed	close	VERB
ejpam-5587	673	13	symmetric	symmetric	ADJ
ejpam-5587	673	14	operators	operator	NOUN
ejpam-5587	673	15	.	.	PUNCT
ejpam-5587	674	1	the	the	DET
ejpam-5587	674	2	initial	initial	ADJ
ejpam-5587	674	3	focus	focus	NOUN
ejpam-5587	674	4	was	be	AUX
ejpam-5587	674	5	on	on	ADP
ejpam-5587	674	6	establishing	establish	VERB
ejpam-5587	674	7	the	the	DET
ejpam-5587	674	8	non	non	NOUN
ejpam-5587	674	9	-	-	NOUN
ejpam-5587	674	10	emptiness	emptiness	NOUN
ejpam-5587	674	11	of	of	ADP
ejpam-5587	674	12	the	the	DET
ejpam-5587	674	13	spectrum	spectrum	NOUN
ejpam-5587	674	14	for	for	ADP
ejpam-5587	674	15	such	such	ADJ
ejpam-5587	674	16	operators	operator	NOUN
ejpam-5587	674	17	,	,	PUNCT
ejpam-5587	674	18	accompanied	accompany	VERB
ejpam-5587	674	19	by	by	ADP
ejpam-5587	674	20	a	a	DET
ejpam-5587	674	21	characterization	characterization	NOUN
ejpam-5587	674	22	of	of	ADP
ejpam-5587	674	23	closed	closed	ADJ
ejpam-5587	674	24	-	-	PUNCT
ejpam-5587	674	25	range	range	NOUN
ejpam-5587	674	26	operators	operator	NOUN
ejpam-5587	674	27	based	base	VERB
ejpam-5587	674	28	on	on	ADP
ejpam-5587	674	29	the	the	DET
ejpam-5587	674	30	spectrum	spectrum	NOUN
ejpam-5587	674	31	.	.	PUNCT
ejpam-5587	675	1	building	build	VERB
ejpam-5587	675	2	on	on	ADP
ejpam-5587	675	3	these	these	DET
ejpam-5587	675	4	foundational	foundational	ADJ
ejpam-5587	675	5	results	result	NOUN
ejpam-5587	675	6	,	,	PUNCT
ejpam-5587	675	7	weyl	weyl	PROPN
ejpam-5587	675	8	’s	’s	PART
ejpam-5587	675	9	theorem	theorem	NOUN
ejpam-5587	675	10	was	be	AUX
ejpam-5587	675	11	proven	prove	VERB
ejpam-5587	675	12	for	for	ADP
ejpam-5587	675	13	densely	densely	ADV
ejpam-5587	675	14	defined	define	VERB
ejpam-5587	675	15	closed	close	VERB
ejpam-5587	675	16	totally	totally	ADV
ejpam-5587	675	17	paranormal	paranormal	ADJ
ejpam-5587	675	18	operators	operator	NOUN
ejpam-5587	675	19	.	.	PUNCT
ejpam-5587	676	1	specifically	specifically	ADV
ejpam-5587	676	2	,	,	PUNCT
ejpam-5587	676	3	it	it	PRON
ejpam-5587	676	4	was	be	AUX
ejpam-5587	676	5	demonstrated	demonstrate	VERB
ejpam-5587	676	6	that	that	SCONJ
ejpam-5587	676	7	the	the	DET
ejpam-5587	676	8	difference	difference	NOUN
ejpam-5587	676	9	between	between	ADP
ejpam-5587	676	10	the	the	DET
ejpam-5587	676	11	spectrum	spectrum	NOUN
ejpam-5587	676	12	σ(t	σ(t	PROPN
ejpam-5587	676	13	)	)	PUNCT
ejpam-5587	676	14	and	and	CCONJ
ejpam-5587	676	15	the	the	DET
ejpam-5587	676	16	weyl	weyl	PROPN
ejpam-5587	676	17	spectrum	spectrum	NOUN
ejpam-5587	676	18	σw(t	σw(t	PUNCT
ejpam-5587	676	19	)	)	PUNCT
ejpam-5587	676	20	is	be	AUX
ejpam-5587	676	21	precisely	precisely	ADV
ejpam-5587	676	22	the	the	DET
ejpam-5587	676	23	set	set	NOUN
ejpam-5587	676	24	of	of	ADP
ejpam-5587	676	25	isolated	isolated	ADJ
ejpam-5587	676	26	eigenvalues	eigenvalue	NOUN
ejpam-5587	676	27	with	with	ADP
ejpam-5587	676	28	finite	finite	PROPN
ejpam-5587	676	29	multiplicities	multiplicity	NOUN
ejpam-5587	676	30	,	,	PUNCT
ejpam-5587	676	31	denoted	denote	VERB
ejpam-5587	676	32	as	as	ADP
ejpam-5587	676	33	π00(t	π00(t	PROPN
ejpam-5587	676	34	)	)	PUNCT
ejpam-5587	676	35	.	.	PUNCT
ejpam-5587	677	1	the	the	DET
ejpam-5587	677	2	final	final	ADJ
ejpam-5587	677	3	section	section	NOUN
ejpam-5587	677	4	of	of	ADP
ejpam-5587	677	5	the	the	DET
ejpam-5587	677	6	article	article	NOUN
ejpam-5587	677	7	explored	explore	VERB
ejpam-5587	677	8	the	the	DET
ejpam-5587	677	9	self	self	NOUN
ejpam-5587	677	10	-	-	PUNCT
ejpam-5587	677	11	adjointness	adjointness	NOUN
ejpam-5587	677	12	of	of	ADP
ejpam-5587	677	13	the	the	DET
ejpam-5587	677	14	riesz	riesz	PROPN
ejpam-5587	677	15	projection	projection	NOUN
ejpam-5587	677	16	eµ	eµ	NOUN
ejpam-5587	677	17	corresponding	correspond	VERB
ejpam-5587	677	18	to	to	ADP
ejpam-5587	677	19	any	any	DET
ejpam-5587	677	20	non	non	ADJ
ejpam-5587	677	21	-	-	ADJ
ejpam-5587	677	22	zero	zero	NUM
ejpam-5587	677	23	isolated	isolate	VERB
ejpam-5587	677	24	spectral	spectral	ADJ
ejpam-5587	677	25	value	value	NOUN
ejpam-5587	677	26	µ	µ	NOUN
ejpam-5587	677	27	of	of	ADP
ejpam-5587	677	28	the	the	DET
ejpam-5587	677	29	operator	operator	NOUN
ejpam-5587	677	30	t	t	NOUN
ejpam-5587	677	31	.	.	PUNCT
ejpam-5587	678	1	the	the	DET
ejpam-5587	678	2	relationships	relationship	NOUN
ejpam-5587	678	3	ran(eµ	ran(eµ	NOUN
ejpam-5587	678	4	)	)	PUNCT
ejpam-5587	678	5	=	=	SYM
ejpam-5587	678	6	ker(t	ker(t	NOUN
ejpam-5587	678	7	−	−	NUM
ejpam-5587	678	8	µi	µi	NOUN
ejpam-5587	678	9	)	)	PUNCT
ejpam-5587	678	10	=	=	PUNCT
ejpam-5587	678	11	ker(t	ker(t	NOUN
ejpam-5587	679	1	−	−	NOUN
ejpam-5587	679	2	µi)∗	µi)∗	NUM
ejpam-5587	679	3	were	be	AUX
ejpam-5587	679	4	established	establish	VERB
ejpam-5587	679	5	for	for	ADP
ejpam-5587	679	6	this	this	DET
ejpam-5587	679	7	riesz	riesz	NOUN
ejpam-5587	679	8	projection	projection	NOUN
ejpam-5587	679	9	.	.	PUNCT
ejpam-5587	680	1	furthermore	furthermore	ADV
ejpam-5587	680	2	,	,	PUNCT
ejpam-5587	680	3	it	it	PRON
ejpam-5587	680	4	was	be	AUX
ejpam-5587	680	5	shown	show	VERB
ejpam-5587	680	6	that	that	SCONJ
ejpam-5587	680	7	if	if	SCONJ
ejpam-5587	680	8	a	a	DET
ejpam-5587	680	9	closed	close	VERB
ejpam-5587	680	10	totally	totally	ADV
ejpam-5587	680	11	paranormal	paranormal	ADJ
ejpam-5587	680	12	operator	operator	NOUN
ejpam-5587	680	13	t	t	PROPN
ejpam-5587	680	14	has	have	VERB
ejpam-5587	680	15	a	a	DET
ejpam-5587	680	16	weyl	weyl	VERB
ejpam-5587	680	17	spectrum	spectrum	NOUN
ejpam-5587	680	18	σw(t	σw(t	PUNCT
ejpam-5587	680	19	)	)	PUNCT
ejpam-5587	680	20	=	=	SYM
ejpam-5587	680	21	0	0	NUM
ejpam-5587	680	22	,	,	PUNCT
ejpam-5587	680	23	then	then	ADV
ejpam-5587	680	24	t	t	PROPN
ejpam-5587	680	25	qualifies	qualify	VERB
ejpam-5587	680	26	as	as	ADP
ejpam-5587	680	27	a	a	DET
ejpam-5587	680	28	compact	compact	ADJ
ejpam-5587	680	29	normal	normal	ADJ
ejpam-5587	680	30	operator	operator	NOUN
ejpam-5587	680	31	.	.	PUNCT
ejpam-5587	681	1	in	in	ADP
ejpam-5587	681	2	terms	term	NOUN
ejpam-5587	681	3	of	of	ADP
ejpam-5587	681	4	future	future	ADJ
ejpam-5587	681	5	work	work	NOUN
ejpam-5587	681	6	,	,	PUNCT
ejpam-5587	681	7	potential	potential	ADJ
ejpam-5587	681	8	avenues	avenue	NOUN
ejpam-5587	681	9	include	include	VERB
ejpam-5587	681	10	exploring	explore	VERB
ejpam-5587	681	11	applications	application	NOUN
ejpam-5587	681	12	of	of	ADP
ejpam-5587	681	13	these	these	DET
ejpam-5587	681	14	spectral	spectral	ADJ
ejpam-5587	681	15	properties	property	NOUN
ejpam-5587	681	16	in	in	ADP
ejpam-5587	681	17	specific	specific	ADJ
ejpam-5587	681	18	mathematical	mathematical	ADJ
ejpam-5587	681	19	or	or	CCONJ
ejpam-5587	681	20	physical	physical	ADJ
ejpam-5587	681	21	contexts	contexts	NOUN
ejpam-5587	681	22	.	.	PUNCT
ejpam-5587	682	1	additionally	additionally	ADV
ejpam-5587	682	2	,	,	PUNCT
ejpam-5587	682	3	investigating	investigate	VERB
ejpam-5587	682	4	the	the	DET
ejpam-5587	682	5	implications	implication	NOUN
ejpam-5587	682	6	of	of	ADP
ejpam-5587	682	7	these	these	DET
ejpam-5587	682	8	results	result	NOUN
ejpam-5587	682	9	on	on	ADP
ejpam-5587	682	10	related	related	ADJ
ejpam-5587	682	11	areas	area	NOUN
ejpam-5587	682	12	of	of	ADP
ejpam-5587	682	13	operator	operator	NOUN
ejpam-5587	682	14	theory	theory	NOUN
ejpam-5587	682	15	or	or	CCONJ
ejpam-5587	682	16	functional	functional	ADJ
ejpam-5587	682	17	analysis	analysis	NOUN
ejpam-5587	682	18	could	could	AUX
ejpam-5587	682	19	provide	provide	VERB
ejpam-5587	682	20	valuable	valuable	ADJ
ejpam-5587	682	21	insights	insight	NOUN
ejpam-5587	682	22	.	.	PUNCT
ejpam-5587	683	1	further	further	ADJ
ejpam-5587	683	2	developments	development	NOUN
ejpam-5587	683	3	in	in	ADP
ejpam-5587	683	4	the	the	DET
ejpam-5587	683	5	understanding	understanding	NOUN
ejpam-5587	683	6	of	of	ADP
ejpam-5587	683	7	totally	totally	ADV
ejpam-5587	683	8	paranormal	paranormal	ADJ
ejpam-5587	683	9	operators	operator	NOUN
ejpam-5587	683	10	and	and	CCONJ
ejpam-5587	683	11	their	their	PRON
ejpam-5587	683	12	spectral	spectral	ADJ
ejpam-5587	683	13	characteristics	characteristic	NOUN
ejpam-5587	683	14	may	may	AUX
ejpam-5587	683	15	contribute	contribute	VERB
ejpam-5587	683	16	to	to	ADP
ejpam-5587	683	17	advancements	advancement	NOUN
ejpam-5587	683	18	in	in	ADP
ejpam-5587	683	19	various	various	ADJ
ejpam-5587	683	20	mathematical	mathematical	ADJ
ejpam-5587	683	21	disciplines	discipline	NOUN
ejpam-5587	683	22	.	.	PUNCT
ejpam-5587	684	1	acknowledgements	acknowledgement	NOUN
ejpam-5587	684	2	the	the	DET
ejpam-5587	684	3	authors	author	NOUN
ejpam-5587	684	4	would	would	AUX
ejpam-5587	684	5	like	like	VERB
ejpam-5587	684	6	to	to	PART
ejpam-5587	684	7	express	express	VERB
ejpam-5587	684	8	our	our	PRON
ejpam-5587	684	9	sincere	sincere	ADJ
ejpam-5587	684	10	gratitude	gratitude	NOUN
ejpam-5587	684	11	to	to	ADP
ejpam-5587	684	12	the	the	DET
ejpam-5587	684	13	referee	referee	NOUN
ejpam-5587	684	14	for	for	ADP
ejpam-5587	684	15	their	their	PRON
ejpam-5587	684	16	valuable	valuable	ADJ
ejpam-5587	684	17	feedback	feedback	NOUN
ejpam-5587	684	18	and	and	CCONJ
ejpam-5587	684	19	insightful	insightful	ADJ
ejpam-5587	684	20	suggestions	suggestion	NOUN
ejpam-5587	684	21	,	,	PUNCT
ejpam-5587	684	22	which	which	PRON
ejpam-5587	684	23	have	have	AUX
ejpam-5587	684	24	significantly	significantly	ADV
ejpam-5587	684	25	contributed	contribute	VERB
ejpam-5587	684	26	to	to	ADP
ejpam-5587	684	27	improving	improve	VERB
ejpam-5587	684	28	the	the	DET
ejpam-5587	684	29	quality	quality	NOUN
ejpam-5587	684	30	and	and	CCONJ
ejpam-5587	684	31	clarity	clarity	NOUN
ejpam-5587	684	32	of	of	ADP
ejpam-5587	684	33	this	this	DET
ejpam-5587	684	34	paper	paper	NOUN
ejpam-5587	684	35	.	.	PUNCT
ejpam-5587	685	1	also	also	ADV
ejpam-5587	685	2	,	,	PUNCT
ejpam-5587	685	3	the	the	DET
ejpam-5587	685	4	first	first	ADJ
ejpam-5587	685	5	author	author	NOUN
ejpam-5587	685	6	extends	extend	VERB
ejpam-5587	685	7	deep	deep	ADJ
ejpam-5587	685	8	gratitude	gratitude	NOUN
ejpam-5587	685	9	to	to	ADP
ejpam-5587	685	10	the	the	DET
ejpam-5587	685	11	deanship	deanship	NOUN
ejpam-5587	685	12	of	of	ADP
ejpam-5587	685	13	scientific	scientific	ADJ
ejpam-5587	685	14	research	research	NOUN
ejpam-5587	685	15	at	at	ADP
ejpam-5587	685	16	the	the	DET
ejpam-5587	685	17	university	university	PROPN
ejpam-5587	685	18	of	of	ADP
ejpam-5587	685	19	jordan	jordan	PROPN
ejpam-5587	685	20	for	for	ADP
ejpam-5587	685	21	their	their	PRON
ejpam-5587	685	22	support	support	NOUN
ejpam-5587	685	23	in	in	ADP
ejpam-5587	685	24	this	this	DET
ejpam-5587	685	25	work	work	NOUN
ejpam-5587	685	26	.	.	PUNCT
ejpam-5587	686	1	s.	s.	PROPN
ejpam-5587	686	2	alnabulsi	alnabulsi	PROPN
ejpam-5587	686	3	,	,	PUNCT
ejpam-5587	686	4	m.h.m	m.h.m	PROPN
ejpam-5587	686	5	.	.	PUNCT
ejpam-5587	686	6	rashid	rashid	PROPN
ejpam-5587	686	7	/	/	SYM
ejpam-5587	686	8	eur	eur	PROPN
ejpam-5587	686	9	.	.	PUNCT
ejpam-5587	687	1	j.	j.	PROPN
ejpam-5587	687	2	pure	pure	PROPN
ejpam-5587	687	3	appl	appl	PROPN
ejpam-5587	687	4	.	.	PROPN
ejpam-5587	687	5	math	math	PROPN
ejpam-5587	687	6	,	,	PUNCT
ejpam-5587	687	7	18	18	NUM
ejpam-5587	687	8	(	(	PUNCT
ejpam-5587	687	9	1	1	NUM
ejpam-5587	687	10	)	)	PUNCT
ejpam-5587	687	11	(	(	PUNCT
ejpam-5587	687	12	2025	2025	NUM
ejpam-5587	687	13	)	)	PUNCT
ejpam-5587	687	14	,	,	PUNCT
ejpam-5587	687	15	5587	5587	NUM
ejpam-5587	687	16	19	19	NUM
ejpam-5587	687	17	of	of	ADP
ejpam-5587	687	18	20	20	NUM
ejpam-5587	687	19	references	reference	NOUN
ejpam-5587	687	20	[	[	X
ejpam-5587	687	21	1	1	NUM
ejpam-5587	687	22	]	]	PUNCT
ejpam-5587	687	23	thomas	thomas	PROPN
ejpam-5587	687	24	n.e	n.e	PROPN
ejpam-5587	687	25	.	.	PROPN
ejpam-5587	687	26	greville	greville	PROPN
ejpam-5587	687	27	adi	adi	PROPN
ejpam-5587	687	28	ben	ben	PROPN
ejpam-5587	687	29	-	-	PROPN
ejpam-5587	687	30	israel	israel	PROPN
ejpam-5587	687	31	.	.	PUNCT
ejpam-5587	688	1	generalized	generalize	VERB
ejpam-5587	688	2	inverses	inverse	NOUN
ejpam-5587	688	3	:	:	PUNCT
ejpam-5587	688	4	theory	theory	NOUN
ejpam-5587	688	5	and	and	CCONJ
ejpam-5587	688	6	applications	application	NOUN
ejpam-5587	688	7	.	.	PUNCT
ejpam-5587	689	1	springer	springer	NOUN
ejpam-5587	689	2	,	,	PUNCT
ejpam-5587	689	3	2	2	NUM
ejpam-5587	689	4	edition	edition	NOUN
ejpam-5587	689	5	,	,	PUNCT
ejpam-5587	689	6	2003	2003	NUM
ejpam-5587	689	7	.	.	PUNCT
ejpam-5587	690	1	[	[	X
ejpam-5587	690	2	2	2	NUM
ejpam-5587	690	3	]	]	X
ejpam-5587	690	4	d.c	d.c	PROPN
ejpam-5587	690	5	.	.	PROPN
ejpam-5587	690	6	lay	lie	VERB
ejpam-5587	690	7	a.e	a.e	PROPN
ejpam-5587	690	8	.	.	PROPN
ejpam-5587	690	9	taylor	taylor	PROPN
ejpam-5587	690	10	.	.	PUNCT
ejpam-5587	691	1	introduction	introduction	NOUN
ejpam-5587	691	2	to	to	ADP
ejpam-5587	691	3	functional	functional	ADJ
ejpam-5587	691	4	analysis	analysis	NOUN
ejpam-5587	691	5	.	.	PUNCT
ejpam-5587	692	1	wiley	wiley	PROPN
ejpam-5587	692	2	,	,	PUNCT
ejpam-5587	692	3	2	2	NUM
ejpam-5587	692	4	edition	edition	NOUN
ejpam-5587	692	5	,	,	PUNCT
ejpam-5587	692	6	1986	1986	NUM
ejpam-5587	692	7	.	.	PUNCT
ejpam-5587	693	1	[	[	X
ejpam-5587	693	2	3	3	X
ejpam-5587	693	3	]	]	PUNCT
ejpam-5587	693	4	t.	t.	PROPN
ejpam-5587	693	5	ando	ando	PROPN
ejpam-5587	693	6	.	.	PUNCT
ejpam-5587	694	1	operators	operator	NOUN
ejpam-5587	694	2	with	with	ADP
ejpam-5587	694	3	a	a	DET
ejpam-5587	694	4	norm	norm	NOUN
ejpam-5587	694	5	condition	condition	NOUN
ejpam-5587	694	6	.	.	PUNCT
ejpam-5587	695	1	acta	acta	PROPN
ejpam-5587	695	2	sci	sci	PROPN
ejpam-5587	695	3	.	.	PROPN
ejpam-5587	695	4	math	math	PROPN
ejpam-5587	695	5	.	.	PUNCT
ejpam-5587	696	1	(	(	PUNCT
ejpam-5587	696	2	szeged	szeged	PROPN
ejpam-5587	696	3	)	)	PUNCT
ejpam-5587	696	4	,	,	PUNCT
ejpam-5587	696	5	33:169–178	33:169–178	NUM
ejpam-5587	696	6	,	,	PUNCT
ejpam-5587	696	7	1972	1972	NUM
ejpam-5587	696	8	.	.	PUNCT
ejpam-5587	697	1	[	[	X
ejpam-5587	697	2	4	4	NUM
ejpam-5587	697	3	]	]	X
ejpam-5587	697	4	neeru	neeru	NOUN
ejpam-5587	697	5	bala	bala	PROPN
ejpam-5587	697	6	and	and	CCONJ
ejpam-5587	697	7	g.	g.	PROPN
ejpam-5587	697	8	ramesh	ramesh	PROPN
ejpam-5587	697	9	.	.	PUNCT
ejpam-5587	698	1	weyl	weyl	PROPN
ejpam-5587	698	2	’s	’s	PART
ejpam-5587	698	3	theorem	theorem	NOUN
ejpam-5587	698	4	for	for	ADP
ejpam-5587	698	5	paranormal	paranormal	ADJ
ejpam-5587	698	6	closed	closed	ADJ
ejpam-5587	698	7	operators	operator	NOUN
ejpam-5587	698	8	.	.	PUNCT
ejpam-5587	699	1	annals	annal	NOUN
ejpam-5587	699	2	of	of	ADP
ejpam-5587	699	3	functional	functional	ADJ
ejpam-5587	699	4	analysis	analysis	NOUN
ejpam-5587	699	5	,	,	PUNCT
ejpam-5587	699	6	11:567	11:567	NUM
ejpam-5587	699	7	–	–	PUNCT
ejpam-5587	699	8	582	582	NUM
ejpam-5587	699	9	,	,	PUNCT
ejpam-5587	699	10	2018	2018	NUM
ejpam-5587	699	11	.	.	PUNCT
ejpam-5587	700	1	[	[	X
ejpam-5587	700	2	5	5	X
ejpam-5587	700	3	]	]	PUNCT
ejpam-5587	700	4	l.	l.	PROPN
ejpam-5587	700	5	coburn	coburn	PROPN
ejpam-5587	700	6	.	.	PUNCT
ejpam-5587	701	1	weyl	weyl	PROPN
ejpam-5587	701	2	’s	’s	PART
ejpam-5587	701	3	theorem	theorem	NOUN
ejpam-5587	701	4	for	for	ADP
ejpam-5587	701	5	nonnormal	nonnormal	ADJ
ejpam-5587	701	6	operators	operator	NOUN
ejpam-5587	701	7	.	.	PUNCT
ejpam-5587	702	1	michigan	michigan	PROPN
ejpam-5587	702	2	math	math	PROPN
ejpam-5587	702	3	.	.	PUNCT
ejpam-5587	703	1	j.	j.	PROPN
ejpam-5587	703	2	,	,	PUNCT
ejpam-5587	703	3	20	20	NUM
ejpam-5587	703	4	,	,	PUNCT
ejpam-5587	703	5	01	01	NUM
ejpam-5587	703	6	1966	1966	NUM
ejpam-5587	703	7	.	.	PUNCT
ejpam-5587	704	1	[	[	X
ejpam-5587	704	2	6	6	NUM
ejpam-5587	704	3	]	]	PUNCT
ejpam-5587	704	4	andrzej	andrzej	PROPN
ejpam-5587	704	5	daniluk	daniluk	PROPN
ejpam-5587	704	6	.	.	PUNCT
ejpam-5587	705	1	on	on	ADP
ejpam-5587	705	2	the	the	DET
ejpam-5587	705	3	closability	closability	NOUN
ejpam-5587	705	4	of	of	ADP
ejpam-5587	705	5	paranormal	paranormal	ADJ
ejpam-5587	705	6	operators	operator	NOUN
ejpam-5587	705	7	.	.	PUNCT
ejpam-5587	706	1	journal	journal	PROPN
ejpam-5587	706	2	of	of	ADP
ejpam-5587	706	3	mathematical	mathematical	ADJ
ejpam-5587	706	4	analysis	analysis	NOUN
ejpam-5587	706	5	and	and	CCONJ
ejpam-5587	706	6	applications	application	NOUN
ejpam-5587	706	7	,	,	PUNCT
ejpam-5587	706	8	376(1):342–348	376(1):342–348	NUM
ejpam-5587	706	9	,	,	PUNCT
ejpam-5587	706	10	2011	2011	NUM
ejpam-5587	706	11	.	.	PUNCT
ejpam-5587	707	1	[	[	X
ejpam-5587	707	2	7	7	X
ejpam-5587	707	3	]	]	X
ejpam-5587	707	4	r.	r.	PROPN
ejpam-5587	707	5	g.	g.	PROPN
ejpam-5587	707	6	douglas	douglas	PROPN
ejpam-5587	707	7	.	.	PUNCT
ejpam-5587	708	1	on	on	ADP
ejpam-5587	708	2	majorization	majorization	NOUN
ejpam-5587	708	3	,	,	PUNCT
ejpam-5587	708	4	factorization	factorization	NOUN
ejpam-5587	708	5	,	,	PUNCT
ejpam-5587	708	6	and	and	CCONJ
ejpam-5587	708	7	range	range	NOUN
ejpam-5587	708	8	inclusion	inclusion	NOUN
ejpam-5587	708	9	of	of	ADP
ejpam-5587	708	10	operators	operator	NOUN
ejpam-5587	708	11	on	on	ADP
ejpam-5587	708	12	hilbert	hilbert	NOUN
ejpam-5587	708	13	space	space	NOUN
ejpam-5587	708	14	.	.	PUNCT
ejpam-5587	709	1	proceedings	proceeding	NOUN
ejpam-5587	709	2	of	of	ADP
ejpam-5587	709	3	the	the	DET
ejpam-5587	709	4	american	american	PROPN
ejpam-5587	709	5	mathematical	mathematical	PROPN
ejpam-5587	709	6	society	society	NOUN
ejpam-5587	709	7	,	,	PUNCT
ejpam-5587	709	8	17(2):413–415	17(2):413–415	NUM
ejpam-5587	709	9	,	,	PUNCT
ejpam-5587	709	10	1966	1966	NUM
ejpam-5587	709	11	.	.	PUNCT
ejpam-5587	710	1	[	[	X
ejpam-5587	710	2	8	8	X
ejpam-5587	710	3	]	]	PUNCT
ejpam-5587	710	4	t.	t.	PROPN
ejpam-5587	710	5	furuta	furuta	PROPN
ejpam-5587	710	6	.	.	PUNCT
ejpam-5587	711	1	on	on	ADP
ejpam-5587	711	2	the	the	DET
ejpam-5587	711	3	class	class	NOUN
ejpam-5587	711	4	of	of	ADP
ejpam-5587	711	5	paranormal	paranormal	ADJ
ejpam-5587	711	6	operators	operator	NOUN
ejpam-5587	711	7	.	.	PUNCT
ejpam-5587	712	1	proceedings	proceeding	NOUN
ejpam-5587	712	2	of	of	ADP
ejpam-5587	712	3	the	the	DET
ejpam-5587	712	4	japan	japan	PROPN
ejpam-5587	712	5	academy	academy	PROPN
ejpam-5587	712	6	,	,	PUNCT
ejpam-5587	712	7	43:594–598	43:594–598	PROPN
ejpam-5587	712	8	,	,	PUNCT
ejpam-5587	712	9	1967	1967	NUM
ejpam-5587	712	10	.	.	PUNCT
ejpam-5587	713	1	[	[	X
ejpam-5587	713	2	9	9	NUM
ejpam-5587	713	3	]	]	X
ejpam-5587	713	4	ramesh	ramesh	PROPN
ejpam-5587	713	5	g	g	PROPN
ejpam-5587	713	6	and	and	CCONJ
ejpam-5587	713	7	s.	s.	PROPN
ejpam-5587	713	8	kulkarni	kulkarni	PROPN
ejpam-5587	713	9	.	.	PUNCT
ejpam-5587	714	1	the	the	DET
ejpam-5587	714	2	carrier	carrier	NOUN
ejpam-5587	714	3	graph	graph	NOUN
ejpam-5587	714	4	topology	topology	NOUN
ejpam-5587	714	5	.	.	PUNCT
ejpam-5587	715	1	banach	banach	PROPN
ejpam-5587	715	2	journal	journal	PROPN
ejpam-5587	715	3	of	of	ADP
ejpam-5587	715	4	mathematical	mathematical	ADJ
ejpam-5587	715	5	analysis	analysis	NOUN
ejpam-5587	715	6	,	,	PUNCT
ejpam-5587	715	7	5:56–69	5:56–69	NUM
ejpam-5587	715	8	,	,	PUNCT
ejpam-5587	715	9	01	01	NUM
ejpam-5587	715	10	2011	2011	NUM
ejpam-5587	715	11	.	.	PUNCT
ejpam-5587	716	1	[	[	X
ejpam-5587	716	2	10	10	NUM
ejpam-5587	716	3	]	]	X
ejpam-5587	716	4	a.	a.	NOUN
ejpam-5587	716	5	gupta	gupta	PROPN
ejpam-5587	716	6	and	and	CCONJ
ejpam-5587	716	7	k.	k.	PROPN
ejpam-5587	716	8	mamtani	mamtani	PROPN
ejpam-5587	716	9	.	.	PUNCT
ejpam-5587	717	1	variants	variant	NOUN
ejpam-5587	717	2	of	of	ADP
ejpam-5587	717	3	weyl	weyl	PROPN
ejpam-5587	717	4	’s	’s	PART
ejpam-5587	717	5	theorem	theorem	NOUN
ejpam-5587	717	6	for	for	ADP
ejpam-5587	717	7	direct	direct	ADJ
ejpam-5587	717	8	sums	sum	NOUN
ejpam-5587	717	9	of	of	ADP
ejpam-5587	717	10	closed	closed	ADJ
ejpam-5587	717	11	linear	linear	PROPN
ejpam-5587	717	12	operators	operator	NOUN
ejpam-5587	717	13	.	.	PUNCT
ejpam-5587	718	1	advances	advance	NOUN
ejpam-5587	718	2	in	in	ADP
ejpam-5587	718	3	operator	operator	NOUN
ejpam-5587	718	4	theory	theory	NOUN
ejpam-5587	718	5	,	,	PUNCT
ejpam-5587	718	6	2:409–418	2:409–418	PROPN
ejpam-5587	718	7	,	,	PUNCT
ejpam-5587	718	8	09	09	NUM
ejpam-5587	718	9	2017	2017	NUM
ejpam-5587	718	10	.	.	PUNCT
ejpam-5587	719	1	[	[	X
ejpam-5587	719	2	11	11	NUM
ejpam-5587	719	3	]	]	X
ejpam-5587	719	4	anuradha	anuradha	PROPN
ejpam-5587	719	5	gupta	gupta	PROPN
ejpam-5587	719	6	and	and	CCONJ
ejpam-5587	719	7	karuna	karuna	PROPN
ejpam-5587	719	8	mamtani	mamtani	PROPN
ejpam-5587	719	9	.	.	PUNCT
ejpam-5587	720	1	weyl	weyl	VERB
ejpam-5587	720	2	type	type	NOUN
ejpam-5587	720	3	theorems	theorem	NOUN
ejpam-5587	720	4	for	for	ADP
ejpam-5587	720	5	unbounded	unbounded	ADJ
ejpam-5587	720	6	hyponormal	hyponormal	ADJ
ejpam-5587	720	7	operators	operator	NOUN
ejpam-5587	720	8	.	.	PUNCT
ejpam-5587	721	1	kyungpook	kyungpook	PROPN
ejpam-5587	721	2	mathematical	mathematical	PROPN
ejpam-5587	721	3	journal	journal	PROPN
ejpam-5587	721	4	,	,	PUNCT
ejpam-5587	721	5	55:531–540	55:531–540	PROPN
ejpam-5587	721	6	,	,	PUNCT
ejpam-5587	721	7	09	09	NUM
ejpam-5587	721	8	2015	2015	NUM
ejpam-5587	721	9	.	.	PUNCT
ejpam-5587	722	1	[	[	X
ejpam-5587	722	2	12	12	NUM
ejpam-5587	722	3	]	]	X
ejpam-5587	722	4	marinus	marinus	PROPN
ejpam-5587	722	5	a.	a.	NOUN
ejpam-5587	722	6	kaashoek	kaashoek	PROPN
ejpam-5587	722	7	israel	israel	PROPN
ejpam-5587	722	8	gohberg	gohberg	PROPN
ejpam-5587	722	9	,	,	PUNCT
ejpam-5587	722	10	seymour	seymour	PROPN
ejpam-5587	722	11	goldberg	goldberg	PROPN
ejpam-5587	722	12	.	.	PUNCT
ejpam-5587	722	13	classes	class	NOUN
ejpam-5587	722	14	of	of	ADP
ejpam-5587	722	15	linear	linear	PROPN
ejpam-5587	722	16	operators	operator	NOUN
ejpam-5587	722	17	,	,	PUNCT
ejpam-5587	722	18	vol	vol	NOUN
ejpam-5587	722	19	.	.	PUNCT
ejpam-5587	723	1	i	i	PRON
ejpam-5587	723	2	,	,	PUNCT
ejpam-5587	723	3	operator	operator	NOUN
ejpam-5587	723	4	theory	theory	NOUN
ejpam-5587	723	5	:	:	PUNCT
ejpam-5587	723	6	advances	advance	NOUN
ejpam-5587	723	7	and	and	CCONJ
ejpam-5587	723	8	applications	application	NOUN
ejpam-5587	723	9	,	,	PUNCT
ejpam-5587	723	10	volume	volume	NOUN
ejpam-5587	723	11	49	49	NUM
ejpam-5587	723	12	.	.	PUNCT
ejpam-5587	724	1	birkhäuser	birkhäuser	X
ejpam-5587	724	2	basel	basel	PROPN
ejpam-5587	724	3	,	,	PUNCT
ejpam-5587	724	4	199	199	NUM
ejpam-5587	724	5	.	.	PUNCT
ejpam-5587	725	1	[	[	X
ejpam-5587	725	2	13	13	NUM
ejpam-5587	725	3	]	]	X
ejpam-5587	725	4	v.	v.	PROPN
ejpam-5587	725	5	istrǎtescu	istrǎtescu	PROPN
ejpam-5587	725	6	.	.	PUNCT
ejpam-5587	726	1	on	on	ADP
ejpam-5587	726	2	some	some	DET
ejpam-5587	726	3	hyponormal	hyponormal	ADJ
ejpam-5587	726	4	operators	operator	NOUN
ejpam-5587	726	5	.	.	PUNCT
ejpam-5587	727	1	pacific	pacific	PROPN
ejpam-5587	727	2	journal	journal	PROPN
ejpam-5587	727	3	of	of	ADP
ejpam-5587	727	4	mathematics	mathematic	NOUN
ejpam-5587	727	5	,	,	PUNCT
ejpam-5587	727	6	22:413–417	22:413–417	NUM
ejpam-5587	727	7	,	,	PUNCT
ejpam-5587	727	8	1967	1967	NUM
ejpam-5587	727	9	.	.	PUNCT
ejpam-5587	728	1	[	[	X
ejpam-5587	728	2	14	14	NUM
ejpam-5587	728	3	]	]	PUNCT
ejpam-5587	728	4	t.	t.	PROPN
ejpam-5587	728	5	kato	kato	PROPN
ejpam-5587	728	6	.	.	PUNCT
ejpam-5587	729	1	perturbation	perturbation	NOUN
ejpam-5587	729	2	theory	theory	NOUN
ejpam-5587	729	3	for	for	ADP
ejpam-5587	729	4	linear	linear	PROPN
ejpam-5587	729	5	operators	operator	NOUN
ejpam-5587	729	6	,	,	PUNCT
ejpam-5587	729	7	volume	volume	NOUN
ejpam-5587	729	8	132	132	NUM
ejpam-5587	729	9	.	.	PUNCT
ejpam-5587	729	10	springer	springer	NOUN
ejpam-5587	729	11	,	,	PUNCT
ejpam-5587	729	12	berlin	berlin	PROPN
ejpam-5587	729	13	,	,	PUNCT
ejpam-5587	729	14	2	2	NUM
ejpam-5587	729	15	edition	edition	NOUN
ejpam-5587	729	16	,	,	PUNCT
ejpam-5587	729	17	198	198	NUM
ejpam-5587	729	18	.	.	PUNCT
ejpam-5587	730	1	[	[	X
ejpam-5587	730	2	15	15	NUM
ejpam-5587	730	3	]	]	X
ejpam-5587	730	4	hideki	hideki	PROPN
ejpam-5587	730	5	kosaki	kosaki	PROPN
ejpam-5587	730	6	.	.	PUNCT
ejpam-5587	731	1	on	on	ADP
ejpam-5587	731	2	intersections	intersection	NOUN
ejpam-5587	731	3	of	of	ADP
ejpam-5587	731	4	domains	domain	NOUN
ejpam-5587	731	5	of	of	ADP
ejpam-5587	731	6	unbounded	unbounded	ADJ
ejpam-5587	731	7	positive	positive	ADJ
ejpam-5587	731	8	operators	operator	NOUN
ejpam-5587	731	9	.	.	PUNCT
ejpam-5587	732	1	kyushu	kyushu	PROPN
ejpam-5587	732	2	journal	journal	PROPN
ejpam-5587	732	3	of	of	ADP
ejpam-5587	732	4	mathematics	mathematic	NOUN
ejpam-5587	732	5	,	,	PUNCT
ejpam-5587	732	6	60	60	NUM
ejpam-5587	732	7	,	,	PUNCT
ejpam-5587	732	8	03	03	NUM
ejpam-5587	732	9	2006	2006	NUM
ejpam-5587	732	10	.	.	PUNCT
ejpam-5587	733	1	[	[	X
ejpam-5587	733	2	16	16	NUM
ejpam-5587	733	3	]	]	PUNCT
ejpam-5587	733	4	s.	s.	PROPN
ejpam-5587	733	5	kulkarni	kulkarni	PROPN
ejpam-5587	733	6	and	and	CCONJ
ejpam-5587	733	7	ramesh	ramesh	PROPN
ejpam-5587	733	8	g.	g.	PROPN
ejpam-5587	733	9	on	on	ADP
ejpam-5587	733	10	the	the	DET
ejpam-5587	733	11	denseness	denseness	NOUN
ejpam-5587	733	12	of	of	ADP
ejpam-5587	733	13	minimum	minimum	NOUN
ejpam-5587	733	14	attaining	attain	VERB
ejpam-5587	733	15	operators	operator	NOUN
ejpam-5587	733	16	.	.	PUNCT
ejpam-5587	734	1	operators	operator	NOUN
ejpam-5587	734	2	and	and	CCONJ
ejpam-5587	734	3	matrices	matrix	NOUN
ejpam-5587	734	4	,	,	PUNCT
ejpam-5587	734	5	12	12	NUM
ejpam-5587	734	6	,	,	PUNCT
ejpam-5587	734	7	09	09	NUM
ejpam-5587	734	8	2016	2016	NUM
ejpam-5587	734	9	.	.	PUNCT
ejpam-5587	735	1	[	[	X
ejpam-5587	735	2	17	17	NUM
ejpam-5587	735	3	]	]	PUNCT
ejpam-5587	735	4	s.	s.	PROPN
ejpam-5587	735	5	kulkarni	kulkarni	PROPN
ejpam-5587	735	6	,	,	PUNCT
ejpam-5587	735	7	m.	m.	PROPN
ejpam-5587	735	8	thamban	thamban	PROPN
ejpam-5587	735	9	nair	nair	PROPN
ejpam-5587	735	10	,	,	PUNCT
ejpam-5587	735	11	and	and	CCONJ
ejpam-5587	735	12	ramesh	ramesh	PROPN
ejpam-5587	735	13	g.	g.	PROPN
ejpam-5587	735	14	some	some	DET
ejpam-5587	735	15	properties	property	NOUN
ejpam-5587	735	16	of	of	ADP
ejpam-5587	735	17	unbounded	unbounded	ADJ
ejpam-5587	735	18	operators	operator	NOUN
ejpam-5587	735	19	with	with	ADP
ejpam-5587	735	20	closed	closed	ADJ
ejpam-5587	735	21	range	range	NOUN
ejpam-5587	735	22	.	.	PUNCT
ejpam-5587	736	1	proceedings	proceeding	NOUN
ejpam-5587	736	2	mathematical	mathematical	ADJ
ejpam-5587	736	3	sciences	science	NOUN
ejpam-5587	736	4	,	,	PUNCT
ejpam-5587	736	5	118:613–625	118:613–625	NUM
ejpam-5587	736	6	,	,	PUNCT
ejpam-5587	736	7	11	11	NUM
ejpam-5587	736	8	2008	2008	NUM
ejpam-5587	736	9	.	.	PUNCT
ejpam-5587	737	1	[	[	X
ejpam-5587	737	2	18	18	NUM
ejpam-5587	737	3	]	]	X
ejpam-5587	737	4	m.t	m.t	PROPN
ejpam-5587	737	5	.	.	PROPN
ejpam-5587	737	6	nair	nair	PROPN
ejpam-5587	737	7	.	.	PUNCT
ejpam-5587	738	1	funcitonal	funcitonal	ADJ
ejpam-5587	738	2	analysis	analysis	NOUN
ejpam-5587	738	3	:	:	PUNCT
ejpam-5587	738	4	a	a	DET
ejpam-5587	738	5	first	first	ADJ
ejpam-5587	738	6	course	course	NOUN
ejpam-5587	738	7	.	.	PUNCT
ejpam-5587	739	1	phi	phi	NOUN
ejpam-5587	739	2	learning	learning	PROPN
ejpam-5587	739	3	pvt	pvt	PROPN
ejpam-5587	739	4	.	.	PROPN
ejpam-5587	739	5	ltd	ltd	PROPN
ejpam-5587	739	6	.	.	PROPN
ejpam-5587	739	7	,	,	PUNCT
ejpam-5587	739	8	2021	2021	NUM
ejpam-5587	739	9	.	.	PUNCT
ejpam-5587	740	1	[	[	X
ejpam-5587	740	2	19	19	NUM
ejpam-5587	740	3	]	]	X
ejpam-5587	740	4	c.	c.	PROPN
ejpam-5587	740	5	r.	r.	PROPN
ejpam-5587	740	6	putnam	putnam	PROPN
ejpam-5587	740	7	.	.	PUNCT
ejpam-5587	741	1	ranges	range	NOUN
ejpam-5587	741	2	of	of	ADP
ejpam-5587	741	3	normal	normal	ADJ
ejpam-5587	741	4	and	and	CCONJ
ejpam-5587	741	5	subnormal	subnormal	ADJ
ejpam-5587	741	6	operators	operator	NOUN
ejpam-5587	741	7	.	.	PUNCT
ejpam-5587	742	1	michigan	michigan	PROPN
ejpam-5587	742	2	mathematical	mathematical	PROPN
ejpam-5587	742	3	journal	journal	PROPN
ejpam-5587	742	4	,	,	PUNCT
ejpam-5587	742	5	18(1):33	18(1):33	NUM
ejpam-5587	742	6	–	–	SYM
ejpam-5587	742	7	36	36	NUM
ejpam-5587	742	8	,	,	PUNCT
ejpam-5587	742	9	1971	1971	NUM
ejpam-5587	742	10	.	.	PUNCT
ejpam-5587	743	1	[	[	X
ejpam-5587	743	2	20	20	NUM
ejpam-5587	743	3	]	]	PUNCT
ejpam-5587	743	4	w.	w.	PROPN
ejpam-5587	743	5	rudin	rudin	PROPN
ejpam-5587	743	6	.	.	PUNCT
ejpam-5587	744	1	functional	functional	ADJ
ejpam-5587	744	2	analysis	analysis	NOUN
ejpam-5587	744	3	.	.	PUNCT
ejpam-5587	745	1	mcgraw	mcgraw	PROPN
ejpam-5587	745	2	-	-	PUNCT
ejpam-5587	745	3	hill	hill	PROPN
ejpam-5587	745	4	,	,	PUNCT
ejpam-5587	745	5	1973	1973	NUM
ejpam-5587	745	6	.	.	PUNCT
ejpam-5587	746	1	[	[	X
ejpam-5587	746	2	21	21	NUM
ejpam-5587	746	3	]	]	PUNCT
ejpam-5587	746	4	m.	m.	NOUN
ejpam-5587	746	5	schechter	schechter	NOUN
ejpam-5587	746	6	.	.	PUNCT
ejpam-5587	747	1	principles	principle	NOUN
ejpam-5587	747	2	of	of	ADP
ejpam-5587	747	3	functional	functional	ADJ
ejpam-5587	747	4	analysis	analysis	NOUN
ejpam-5587	747	5	.	.	PUNCT
ejpam-5587	748	1	american	american	PROPN
ejpam-5587	748	2	mathematical	mathematical	PROPN
ejpam-5587	748	3	society	society	NOUN
ejpam-5587	748	4	,	,	PUNCT
ejpam-5587	748	5	2	2	NUM
ejpam-5587	748	6	edition	edition	NOUN
ejpam-5587	748	7	,	,	PUNCT
ejpam-5587	748	8	1971	1971	NUM
ejpam-5587	748	9	.	.	PUNCT
ejpam-5587	749	1	s.	s.	PROPN
ejpam-5587	749	2	alnabulsi	alnabulsi	PROPN
ejpam-5587	749	3	,	,	PUNCT
ejpam-5587	749	4	m.h.m	m.h.m	PROPN
ejpam-5587	749	5	.	.	PUNCT
ejpam-5587	749	6	rashid	rashid	PROPN
ejpam-5587	749	7	/	/	SYM
ejpam-5587	749	8	eur	eur	PROPN
ejpam-5587	749	9	.	.	PUNCT
ejpam-5587	750	1	j.	j.	PROPN
ejpam-5587	750	2	pure	pure	PROPN
ejpam-5587	750	3	appl	appl	PROPN
ejpam-5587	750	4	.	.	PROPN
ejpam-5587	750	5	math	math	PROPN
ejpam-5587	750	6	,	,	PUNCT
ejpam-5587	750	7	18	18	NUM
ejpam-5587	750	8	(	(	PUNCT
ejpam-5587	750	9	1	1	NUM
ejpam-5587	750	10	)	)	PUNCT
ejpam-5587	750	11	(	(	PUNCT
ejpam-5587	750	12	2025	2025	NUM
ejpam-5587	750	13	)	)	PUNCT
ejpam-5587	750	14	,	,	PUNCT
ejpam-5587	750	15	5587	5587	NUM
ejpam-5587	750	16	20	20	NUM
ejpam-5587	750	17	of	of	ADP
ejpam-5587	750	18	20	20	NUM
ejpam-5587	750	19	[	[	SYM
ejpam-5587	750	20	22	22	NUM
ejpam-5587	750	21	]	]	PUNCT
ejpam-5587	750	22	c.	c.	NOUN
ejpam-5587	750	23	schmoeger	schmoeger	NOUN
ejpam-5587	750	24	.	.	PUNCT
ejpam-5587	751	1	on	on	ADP
ejpam-5587	751	2	totally	totally	ADV
ejpam-5587	751	3	paranormal	paranormal	ADJ
ejpam-5587	751	4	operators	operator	NOUN
ejpam-5587	751	5	.	.	PUNCT
ejpam-5587	752	1	bull	bull	NOUN
ejpam-5587	752	2	.	.	PUNCT
ejpam-5587	753	1	austral	austral	PROPN
ejpam-5587	753	2	.	.	PUNCT
ejpam-5587	753	3	math	math	NOUN
ejpam-5587	753	4	.	.	PUNCT
ejpam-5587	754	1	soc	soc	PROPN
ejpam-5587	754	2	.	.	PUNCT
ejpam-5587	754	3	,	,	PUNCT
ejpam-5587	754	4	66(3):425	66(3):425	PROPN
ejpam-5587	754	5	–	–	PUNCT
ejpam-5587	754	6	441	441	NUM
ejpam-5587	754	7	,	,	PUNCT
ejpam-5587	754	8	2002	2002	NUM
ejpam-5587	754	9	.	.	PUNCT
ejpam-5587	755	1	[	[	X
ejpam-5587	755	2	23	23	NUM
ejpam-5587	755	3	]	]	PUNCT
ejpam-5587	755	4	jan	jan	PROPN
ejpam-5587	755	5	stochel	stochel	PROPN
ejpam-5587	755	6	.	.	PUNCT
ejpam-5587	756	1	an	an	DET
ejpam-5587	756	2	asymmetric	asymmetric	ADJ
ejpam-5587	756	3	putnam	putnam	PROPN
ejpam-5587	756	4	–	–	PUNCT
ejpam-5587	756	5	fuglede	fuglede	PROPN
ejpam-5587	756	6	theorem	theorem	NOUN
ejpam-5587	756	7	for	for	ADP
ejpam-5587	756	8	unbounded	unbounded	ADJ
ejpam-5587	756	9	operators	operator	NOUN
ejpam-5587	756	10	.	.	PUNCT
ejpam-5587	757	1	proceedings	proceeding	NOUN
ejpam-5587	757	2	of	of	ADP
ejpam-5587	757	3	the	the	DET
ejpam-5587	757	4	american	american	PROPN
ejpam-5587	757	5	mathematical	mathematical	PROPN
ejpam-5587	757	6	society	society	NOUN
ejpam-5587	757	7	,	,	PUNCT
ejpam-5587	757	8	129:2261–2271	129:2261–2271	NUM
ejpam-5587	757	9	,	,	PUNCT
ejpam-5587	757	10	03	03	NUM
ejpam-5587	757	11	2001	2001	NUM
ejpam-5587	757	12	.	.	PUNCT
ejpam-5587	758	1	[	[	X
ejpam-5587	758	2	24	24	NUM
ejpam-5587	758	3	]	]	X
ejpam-5587	758	4	schǒichi	schǒichi	PROPN
ejpam-5587	758	5	ǒta	ǒta	PROPN
ejpam-5587	758	6	and	and	CCONJ
ejpam-5587	758	7	konrad	konrad	PROPN
ejpam-5587	758	8	schmüdgen	schmüdgen	PROPN
ejpam-5587	758	9	.	.	PUNCT
ejpam-5587	759	1	some	some	DET
ejpam-5587	759	2	classes	class	NOUN
ejpam-5587	759	3	of	of	ADP
ejpam-5587	759	4	unbounded	unbounded	ADJ
ejpam-5587	759	5	operators	operator	NOUN
ejpam-5587	759	6	.	.	PUNCT
ejpam-5587	760	1	integral	integral	ADJ
ejpam-5587	760	2	equations	equation	NOUN
ejpam-5587	760	3	and	and	CCONJ
ejpam-5587	760	4	operator	operator	NOUN
ejpam-5587	760	5	theory	theory	NOUN
ejpam-5587	760	6	,	,	PUNCT
ejpam-5587	760	7	12:211–226	12:211–226	PROPN
ejpam-5587	760	8	,	,	PUNCT
ejpam-5587	760	9	03	03	NUM
ejpam-5587	760	10	1989	1989	NUM
ejpam-5587	760	11	.	.	PUNCT
ejpam-5587	761	1	[	[	X
ejpam-5587	761	2	25	25	NUM
ejpam-5587	761	3	]	]	X
ejpam-5587	761	4	r.	r.	PROPN
ejpam-5587	761	5	walter	walter	PROPN
ejpam-5587	761	6	.	.	PUNCT
ejpam-5587	762	1	functional	functional	ADJ
ejpam-5587	762	2	analysis	analysis	NOUN
ejpam-5587	762	3	:	:	PUNCT
ejpam-5587	762	4	international	international	ADJ
ejpam-5587	762	5	series	series	NOUN
ejpam-5587	762	6	in	in	ADP
ejpam-5587	762	7	pure	pure	ADJ
ejpam-5587	762	8	and	and	CCONJ
ejpam-5587	762	9	applied	applied	ADJ
ejpam-5587	762	10	mathematics	mathematic	NOUN
ejpam-5587	762	11	.	.	PUNCT
ejpam-5587	763	1	new	new	PROPN
ejpam-5587	763	2	york	york	PROPN
ejpam-5587	763	3	:	:	PUNCT
ejpam-5587	763	4	mcgraw	mcgraw	PROPN
ejpam-5587	763	5	-	-	PUNCT
ejpam-5587	763	6	hill	hill	PROPN
ejpam-5587	763	7	inc	inc	PROPN
ejpam-5587	763	8	.	.	PROPN
ejpam-5587	763	9	,	,	PUNCT
ejpam-5587	763	10	2	2	NUM
ejpam-5587	763	11	edition	edition	NOUN
ejpam-5587	763	12	,	,	PUNCT
ejpam-5587	763	13	1991	1991	NUM
ejpam-5587	763	14	.	.	PUNCT