id	sid	tid	token	lemma	pos
ap-1412	1	1	acta	acta	PROPN
ap-1412	1	2	polytechnica	polytechnica	PROPN
ap-1412	1	3	vol	vol	NOUN
ap-1412	1	4	.	.	PUNCT
ap-1412	2	1	51	51	NUM
ap-1412	2	2	no	no	INTJ
ap-1412	2	3	.	.	PUNCT
ap-1412	3	1	4/2011	4/2011	NUM
ap-1412	3	2	effect	effect	NOUN
ap-1412	3	3	algebras	algebra	NOUN
ap-1412	3	4	of	of	ADP
ap-1412	3	5	positive	positive	ADJ
ap-1412	3	6	self	self	NOUN
ap-1412	3	7	-	-	PUNCT
ap-1412	3	8	adjoint	adjoint	NOUN
ap-1412	3	9	operators	operator	NOUN
ap-1412	3	10	densely	densely	ADV
ap-1412	3	11	defined	define	VERB
ap-1412	3	12	on	on	ADP
ap-1412	3	13	hilbert	hilbert	PROPN
ap-1412	3	14	spaces	space	NOUN
ap-1412	3	15	z.	z.	PROPN
ap-1412	3	16	riečanová	riečanová	PROPN
ap-1412	4	1	abstract	abstract	ADV
ap-1412	4	2	we	we	PRON
ap-1412	4	3	show	show	VERB
ap-1412	4	4	that	that	SCONJ
ap-1412	4	5	(	(	PUNCT
ap-1412	4	6	generalized	generalized	ADJ
ap-1412	4	7	)	)	PUNCT
ap-1412	4	8	effect	effect	NOUN
ap-1412	4	9	algebras	algebra	NOUN
ap-1412	4	10	may	may	AUX
ap-1412	4	11	be	be	AUX
ap-1412	4	12	suitable	suitable	ADJ
ap-1412	4	13	very	very	ADV
ap-1412	4	14	simple	simple	ADJ
ap-1412	4	15	and	and	CCONJ
ap-1412	4	16	natural	natural	ADJ
ap-1412	4	17	algebraic	algebraic	ADJ
ap-1412	4	18	structures	structure	NOUN
ap-1412	4	19	for	for	ADP
ap-1412	4	20	sets	set	NOUN
ap-1412	4	21	of	of	ADP
ap-1412	4	22	(	(	PUNCT
ap-1412	4	23	unbounded	unbounded	ADJ
ap-1412	4	24	)	)	PUNCT
ap-1412	4	25	positive	positive	ADJ
ap-1412	4	26	self	self	NOUN
ap-1412	4	27	-	-	PUNCT
ap-1412	4	28	adjoint	adjoint	NOUN
ap-1412	4	29	linear	linear	PROPN
ap-1412	4	30	operators	operator	NOUN
ap-1412	4	31	densely	densely	ADV
ap-1412	4	32	defined	define	VERB
ap-1412	4	33	on	on	ADP
ap-1412	4	34	an	an	DET
ap-1412	4	35	infinite	infinite	ADJ
ap-1412	4	36	-	-	PUNCT
ap-1412	4	37	dimensional	dimensional	ADJ
ap-1412	4	38	complex	complex	ADJ
ap-1412	4	39	hilbert	hilbert	NOUN
ap-1412	4	40	space	space	NOUN
ap-1412	4	41	.	.	PUNCT
ap-1412	5	1	in	in	ADP
ap-1412	5	2	these	these	DET
ap-1412	5	3	cases	case	NOUN
ap-1412	5	4	the	the	DET
ap-1412	5	5	effect	effect	NOUN
ap-1412	5	6	algebraic	algebraic	ADJ
ap-1412	5	7	operation	operation	NOUN
ap-1412	5	8	,	,	PUNCT
ap-1412	5	9	as	as	ADP
ap-1412	5	10	a	a	DET
ap-1412	5	11	total	total	NOUN
ap-1412	5	12	or	or	CCONJ
ap-1412	5	13	partially	partially	ADV
ap-1412	5	14	defined	define	VERB
ap-1412	5	15	binary	binary	ADJ
ap-1412	5	16	operation	operation	NOUN
ap-1412	5	17	,	,	PUNCT
ap-1412	5	18	coincides	coincide	VERB
ap-1412	5	19	with	with	ADP
ap-1412	5	20	the	the	DET
ap-1412	5	21	usual	usual	ADJ
ap-1412	5	22	addition	addition	NOUN
ap-1412	5	23	of	of	ADP
ap-1412	5	24	operators	operator	NOUN
ap-1412	5	25	in	in	ADP
ap-1412	5	26	hilbert	hilbert	PROPN
ap-1412	5	27	spaces	space	NOUN
ap-1412	5	28	.	.	PUNCT
ap-1412	6	1	keywords	keyword	NOUN
ap-1412	6	2	:	:	PUNCT
ap-1412	6	3	quantum	quantum	ADJ
ap-1412	6	4	structures	structure	NOUN
ap-1412	6	5	,	,	PUNCT
ap-1412	6	6	(	(	PUNCT
ap-1412	6	7	generalized	generalized	ADJ
ap-1412	6	8	)	)	PUNCT
ap-1412	6	9	effect	effect	NOUN
ap-1412	6	10	algebra	algebra	NOUN
ap-1412	6	11	,	,	PUNCT
ap-1412	6	12	hilbert	hilbert	NOUN
ap-1412	6	13	space	space	NOUN
ap-1412	6	14	,	,	PUNCT
ap-1412	6	15	(	(	PUNCT
ap-1412	6	16	unbounded	unbounded	ADJ
ap-1412	6	17	)	)	PUNCT
ap-1412	6	18	positive	positive	ADJ
ap-1412	6	19	linear	linear	NOUN
ap-1412	6	20	operator	operator	NOUN
ap-1412	6	21	.	.	PUNCT
ap-1412	7	1	1	1	NUM
ap-1412	7	2	introduction	introduction	NOUN
ap-1412	7	3	for	for	ADP
ap-1412	7	4	any	any	DET
ap-1412	7	5	linear	linear	ADJ
ap-1412	7	6	operator	operator	NOUN
ap-1412	7	7	a	a	PRON
ap-1412	7	8	densely	densely	ADV
ap-1412	7	9	defined	define	VERB
ap-1412	7	10	on	on	ADP
ap-1412	7	11	a	a	DET
ap-1412	7	12	hilbert	hilbert	NOUN
ap-1412	7	13	space	space	NOUN
ap-1412	7	14	h	h	NOUN
ap-1412	7	15	one	one	PRON
ap-1412	7	16	can	can	AUX
ap-1412	7	17	define	define	VERB
ap-1412	7	18	its	its	PRON
ap-1412	7	19	adjoint	adjoint	NOUN
ap-1412	7	20	operator	operator	NOUN
ap-1412	7	21	a∗.	a∗.	NOUN
ap-1412	7	22	if	if	SCONJ
ap-1412	7	23	a∗	a∗	PROPN
ap-1412	7	24	coincides	coincide	VERB
ap-1412	7	25	with	with	ADP
ap-1412	7	26	a	a	DET
ap-1412	7	27	then	then	ADV
ap-1412	7	28	operator	operator	NOUN
ap-1412	7	29	a	a	PRON
ap-1412	7	30	is	be	AUX
ap-1412	7	31	called	call	VERB
ap-1412	7	32	selfadjoint	selfadjoint	NOUN
ap-1412	7	33	.	.	PUNCT
ap-1412	8	1	self	self	NOUN
ap-1412	8	2	-	-	PUNCT
ap-1412	8	3	adjoint	adjoint	NOUN
ap-1412	8	4	(	(	PUNCT
ap-1412	8	5	unbounded	unbounded	ADJ
ap-1412	8	6	)	)	PUNCT
ap-1412	8	7	linear	linear	PROPN
ap-1412	8	8	operators	operator	NOUN
ap-1412	8	9	on	on	ADP
ap-1412	8	10	infinite	infinite	ADJ
ap-1412	8	11	-	-	PUNCT
ap-1412	8	12	dimensional	dimensional	ADJ
ap-1412	8	13	complex	complex	ADJ
ap-1412	8	14	hilbert	hilbert	NOUN
ap-1412	8	15	spaces	space	NOUN
ap-1412	8	16	have	have	VERB
ap-1412	8	17	importance	importance	NOUN
ap-1412	8	18	in	in	ADP
ap-1412	8	19	quantum	quantum	ADJ
ap-1412	8	20	mechanics	mechanic	NOUN
ap-1412	8	21	,	,	PUNCT
ap-1412	8	22	since	since	SCONJ
ap-1412	8	23	they	they	PRON
ap-1412	8	24	represent	represent	VERB
ap-1412	8	25	physical	physical	ADJ
ap-1412	8	26	observables	observable	NOUN
ap-1412	8	27	,	,	PUNCT
ap-1412	8	28	e.g.	e.g.	ADV
ap-1412	8	29	the	the	DET
ap-1412	8	30	position	position	NOUN
ap-1412	8	31	or	or	CCONJ
ap-1412	8	32	momentum	momentum	NOUN
ap-1412	8	33	of	of	ADP
ap-1412	8	34	an	an	DET
ap-1412	8	35	elementary	elementary	ADJ
ap-1412	8	36	particle	particle	NOUN
ap-1412	8	37	.	.	PUNCT
ap-1412	9	1	differential	differential	PROPN
ap-1412	9	2	operators	operator	NOUN
ap-1412	9	3	form	form	VERB
ap-1412	9	4	a	a	DET
ap-1412	9	5	class	class	NOUN
ap-1412	9	6	of	of	ADP
ap-1412	9	7	unbounded	unbounded	ADJ
ap-1412	9	8	operators	operator	NOUN
ap-1412	9	9	.	.	PUNCT
ap-1412	10	1	the	the	DET
ap-1412	10	2	laplace	laplace	NOUN
ap-1412	10	3	operator	operator	NOUN
ap-1412	10	4	is	be	AUX
ap-1412	10	5	an	an	DET
ap-1412	10	6	example	example	NOUN
ap-1412	10	7	of	of	ADP
ap-1412	10	8	an	an	DET
ap-1412	10	9	unbounded	unbounded	ADJ
ap-1412	10	10	positive	positive	ADJ
ap-1412	10	11	linear	linear	NOUN
ap-1412	10	12	operator	operator	NOUN
ap-1412	10	13	.	.	PUNCT
ap-1412	11	1	the	the	DET
ap-1412	11	2	algebraic	algebraic	ADJ
ap-1412	11	3	structures	structure	NOUN
ap-1412	11	4	of	of	ADP
ap-1412	11	5	sets	set	NOUN
ap-1412	11	6	of	of	ADP
ap-1412	11	7	such	such	ADJ
ap-1412	11	8	operators	operator	NOUN
ap-1412	11	9	distinguish	distinguish	VERB
ap-1412	11	10	from	from	ADP
ap-1412	11	11	classical	classical	ADJ
ap-1412	11	12	boolean	boolean	ADJ
ap-1412	11	13	logics	logic	NOUN
ap-1412	11	14	.	.	PUNCT
ap-1412	12	1	this	this	PRON
ap-1412	12	2	follows	follow	VERB
ap-1412	12	3	from	from	ADP
ap-1412	12	4	the	the	DET
ap-1412	12	5	fact	fact	NOUN
ap-1412	12	6	that	that	SCONJ
ap-1412	12	7	e.g.	e.g.	ADV
ap-1412	12	8	,	,	PUNCT
ap-1412	12	9	the	the	DET
ap-1412	12	10	distributive	distributive	ADJ
ap-1412	12	11	law	law	NOUN
ap-1412	12	12	fails	fail	VERB
ap-1412	12	13	due	due	ADP
ap-1412	12	14	to	to	ADP
ap-1412	12	15	the	the	DET
ap-1412	12	16	noncompatibility	noncompatibility	NOUN
ap-1412	12	17	of	of	ADP
ap-1412	12	18	some	some	DET
ap-1412	12	19	pairs	pair	NOUN
ap-1412	12	20	of	of	ADP
ap-1412	12	21	operators	operator	NOUN
ap-1412	12	22	.	.	PUNCT
ap-1412	13	1	for	for	ADP
ap-1412	13	2	instance	instance	NOUN
ap-1412	13	3	,	,	PUNCT
ap-1412	13	4	position	position	NOUN
ap-1412	13	5	x	x	PUNCT
ap-1412	13	6	and	and	CCONJ
ap-1412	13	7	momentum	momentum	NOUN
ap-1412	13	8	p	p	NOUN
ap-1412	13	9	of	of	ADP
ap-1412	13	10	an	an	DET
ap-1412	13	11	elementary	elementary	ADJ
ap-1412	13	12	particle	particle	NOUN
ap-1412	13	13	can	can	AUX
ap-1412	13	14	not	not	PART
ap-1412	13	15	be	be	AUX
ap-1412	13	16	measurable	measurable	ADJ
ap-1412	13	17	simultaneously	simultaneously	ADV
ap-1412	13	18	with	with	ADP
ap-1412	13	19	arbitrarily	arbitrarily	ADV
ap-1412	13	20	prescribed	prescribe	VERB
ap-1412	13	21	accuracy	accuracy	NOUN
ap-1412	13	22	,	,	PUNCT
ap-1412	13	23	hence	hence	ADV
ap-1412	13	24	x	x	X
ap-1412	13	25	and	and	CCONJ
ap-1412	13	26	p	p	NOUN
ap-1412	13	27	are	be	AUX
ap-1412	13	28	noncompatible	noncompatible	ADJ
ap-1412	13	29	.	.	PUNCT
ap-1412	14	1	non	non	ADJ
ap-1412	14	2	-	-	ADJ
ap-1412	14	3	classical	classical	ADJ
ap-1412	14	4	logic	logic	NOUN
ap-1412	14	5	for	for	ADP
ap-1412	14	6	calculus	calculus	NOUN
ap-1412	14	7	of	of	ADP
ap-1412	14	8	propositions	proposition	NOUN
ap-1412	14	9	of	of	ADP
ap-1412	14	10	quantum	quantum	ADJ
ap-1412	14	11	mechanical	mechanical	ADJ
ap-1412	14	12	system	system	NOUN
ap-1412	14	13	started	start	VERB
ap-1412	14	14	in	in	ADP
ap-1412	14	15	1936	1936	NUM
ap-1412	14	16	by	by	ADP
ap-1412	14	17	birkhoff	birkhoff	NOUN
ap-1412	14	18	and	and	CCONJ
ap-1412	14	19	von	von	PROPN
ap-1412	14	20	neuman	neuman	PROPN
ap-1412	14	21	,	,	PUNCT
ap-1412	14	22	(	(	PUNCT
ap-1412	14	23	see	see	VERB
ap-1412	14	24	[	[	X
ap-1412	14	25	2	2	NUM
ap-1412	14	26	]	]	PUNCT
ap-1412	14	27	)	)	PUNCT
ap-1412	14	28	.	.	PUNCT
ap-1412	15	1	effect	effect	NOUN
ap-1412	15	2	algebras	algebra	NOUN
ap-1412	15	3	were	be	AUX
ap-1412	15	4	introduced	introduce	VERB
ap-1412	15	5	in	in	ADP
ap-1412	15	6	1994	1994	NUM
ap-1412	15	7	in	in	ADP
ap-1412	15	8	[	[	X
ap-1412	15	9	5	5	NUM
ap-1412	15	10	]	]	PUNCT
ap-1412	15	11	.	.	PUNCT
ap-1412	16	1	a	a	DET
ap-1412	16	2	survey	survey	NOUN
ap-1412	16	3	of	of	ADP
ap-1412	16	4	algebras	algebra	NOUN
ap-1412	16	5	of	of	ADP
ap-1412	16	6	unbounded	unbounded	ADJ
ap-1412	16	7	operators	operator	NOUN
ap-1412	16	8	can	can	AUX
ap-1412	16	9	be	be	AUX
ap-1412	16	10	found	find	VERB
ap-1412	16	11	in	in	ADP
ap-1412	16	12	[	[	X
ap-1412	16	13	1	1	NUM
ap-1412	16	14	]	]	PUNCT
ap-1412	16	15	.	.	PUNCT
ap-1412	17	1	the	the	DET
ap-1412	17	2	aim	aim	NOUN
ap-1412	17	3	of	of	ADP
ap-1412	17	4	this	this	DET
ap-1412	17	5	paper	paper	NOUN
ap-1412	17	6	is	be	AUX
ap-1412	17	7	to	to	PART
ap-1412	17	8	show	show	VERB
ap-1412	17	9	that	that	SCONJ
ap-1412	17	10	(	(	PUNCT
ap-1412	17	11	generalized	generalized	ADJ
ap-1412	17	12	)	)	PUNCT
ap-1412	17	13	effect	effect	NOUN
ap-1412	17	14	algebras	algebra	NOUN
ap-1412	17	15	may	may	AUX
ap-1412	17	16	be	be	AUX
ap-1412	17	17	suitable	suitable	ADJ
ap-1412	17	18	,	,	PUNCT
ap-1412	17	19	very	very	ADV
ap-1412	17	20	simple	simple	ADJ
ap-1412	17	21	and	and	CCONJ
ap-1412	17	22	natural	natural	ADJ
ap-1412	17	23	algebraic	algebraic	ADJ
ap-1412	17	24	structures	structure	NOUN
ap-1412	17	25	for	for	ADP
ap-1412	17	26	sets	set	NOUN
ap-1412	17	27	of	of	ADP
ap-1412	17	28	linear	linear	PROPN
ap-1412	17	29	operators	operator	NOUN
ap-1412	17	30	(	(	PUNCT
ap-1412	17	31	including	include	VERB
ap-1412	17	32	unbounded	unbounded	ADJ
ap-1412	17	33	ones	one	NOUN
ap-1412	17	34	)	)	PUNCT
ap-1412	17	35	densely	densely	ADV
ap-1412	17	36	defined	define	VERB
ap-1412	17	37	on	on	ADP
ap-1412	17	38	an	an	DET
ap-1412	17	39	infinite	infinite	ADJ
ap-1412	17	40	-	-	PUNCT
ap-1412	17	41	dimensional	dimensional	ADJ
ap-1412	17	42	complex	complex	ADJ
ap-1412	17	43	hilbert	hilbert	NOUN
ap-1412	17	44	space	space	NOUN
ap-1412	17	45	,	,	PUNCT
ap-1412	17	46	at	at	ADP
ap-1412	17	47	which	which	PRON
ap-1412	17	48	the	the	DET
ap-1412	17	49	effect	effect	NOUN
ap-1412	17	50	algebraic	algebraic	ADJ
ap-1412	17	51	operation	operation	NOUN
ap-1412	17	52	coincides	coincide	VERB
ap-1412	17	53	with	with	ADP
ap-1412	17	54	the	the	DET
ap-1412	17	55	usual	usual	ADJ
ap-1412	17	56	sum	sum	NOUN
ap-1412	17	57	of	of	ADP
ap-1412	17	58	operators	operator	NOUN
ap-1412	17	59	.	.	PUNCT
ap-1412	18	1	more	more	ADJ
ap-1412	18	2	details	detail	NOUN
ap-1412	18	3	on	on	ADP
ap-1412	18	4	linear	linear	PROPN
ap-1412	18	5	operators	operator	NOUN
ap-1412	18	6	on	on	ADP
ap-1412	18	7	hilbert	hilbert	NOUN
ap-1412	18	8	spaces	space	NOUN
ap-1412	18	9	can	can	AUX
ap-1412	18	10	be	be	AUX
ap-1412	18	11	found	find	VERB
ap-1412	18	12	,	,	PUNCT
ap-1412	18	13	e.g.	e.g.	ADV
ap-1412	18	14	,	,	PUNCT
ap-1412	18	15	in	in	ADP
ap-1412	18	16	[	[	X
ap-1412	18	17	3	3	NUM
ap-1412	18	18	]	]	PUNCT
ap-1412	18	19	and	and	CCONJ
ap-1412	18	20	about	about	ADP
ap-1412	18	21	effect	effect	NOUN
ap-1412	18	22	algebras	algebra	VERB
ap-1412	18	23	in	in	ADP
ap-1412	18	24	[	[	X
ap-1412	18	25	4	4	NUM
ap-1412	18	26	]	]	PUNCT
ap-1412	18	27	.	.	PUNCT
ap-1412	19	1	2	2	NUM
ap-1412	19	2	basic	basic	ADJ
ap-1412	19	3	definitions	definition	NOUN
ap-1412	19	4	and	and	CCONJ
ap-1412	19	5	some	some	DET
ap-1412	19	6	known	know	VERB
ap-1412	19	7	facts	fact	NOUN
ap-1412	19	8	in	in	ADP
ap-1412	19	9	the	the	DET
ap-1412	19	10	paper	paper	NOUN
ap-1412	19	11	we	we	PRON
ap-1412	19	12	assume	assume	VERB
ap-1412	19	13	that	that	SCONJ
ap-1412	19	14	h	h	NOUN
ap-1412	19	15	is	be	AUX
ap-1412	19	16	an	an	DET
ap-1412	19	17	infinitedimensional	infinitedimensional	ADJ
ap-1412	19	18	complex	complex	ADJ
ap-1412	19	19	hilbert	hilbert	NOUN
ap-1412	19	20	space	space	NOUN
ap-1412	19	21	,	,	PUNCT
ap-1412	19	22	i.e.	i.e.	X
ap-1412	19	23	,	,	PUNCT
ap-1412	19	24	a	a	DET
ap-1412	19	25	linear	linear	ADJ
ap-1412	19	26	space	space	NOUN
ap-1412	19	27	with	with	ADP
ap-1412	19	28	inner	inner	ADJ
ap-1412	19	29	product	product	NOUN
ap-1412	19	30	(	(	PUNCT
ap-1412	19	31	·	·	PUNCT
ap-1412	19	32	,	,	PUNCT
ap-1412	19	33	·	·	PUNCT
ap-1412	19	34	)	)	PUNCT
ap-1412	19	35	which	which	PRON
ap-1412	19	36	is	be	AUX
ap-1412	19	37	complete	complete	ADJ
ap-1412	19	38	in	in	ADP
ap-1412	19	39	the	the	DET
ap-1412	19	40	induced	induced	ADJ
ap-1412	19	41	metric	metric	NOUN
ap-1412	19	42	.	.	PUNCT
ap-1412	20	1	conventions	convention	NOUN
ap-1412	20	2	differ	differ	VERB
ap-1412	20	3	as	as	ADP
ap-1412	20	4	to	to	PART
ap-1412	20	5	which	which	DET
ap-1412	20	6	argument	argument	NOUN
ap-1412	20	7	sesquilinear	sesquilinear	NOUN
ap-1412	20	8	form	form	NOUN
ap-1412	20	9	(	(	PUNCT
ap-1412	20	10	·	·	PUNCT
ap-1412	20	11	,	,	PUNCT
ap-1412	20	12	·	·	PUNCT
ap-1412	20	13	)	)	PUNCT
ap-1412	20	14	should	should	AUX
ap-1412	20	15	be	be	AUX
ap-1412	20	16	linear	linear	ADJ
ap-1412	20	17	.	.	PUNCT
ap-1412	21	1	recall	recall	VERB
ap-1412	21	2	that	that	PRON
ap-1412	21	3	here	here	ADV
ap-1412	21	4	for	for	ADP
ap-1412	21	5	any	any	DET
ap-1412	21	6	x	x	NOUN
ap-1412	21	7	,	,	PUNCT
ap-1412	21	8	y	y	PROPN
ap-1412	21	9	∈	∈	PROPN
ap-1412	21	10	h	h	NOUN
ap-1412	21	11	we	we	PRON
ap-1412	21	12	have	have	VERB
ap-1412	21	13	(	(	PUNCT
ap-1412	22	1	x	x	NOUN
ap-1412	22	2	,	,	PUNCT
ap-1412	22	3	y	y	NOUN
ap-1412	22	4	)	)	PUNCT
ap-1412	22	5	∈	∈	PROPN
ap-1412	22	6	c	c	NOUN
ap-1412	22	7	(	(	PUNCT
ap-1412	22	8	the	the	DET
ap-1412	22	9	set	set	NOUN
ap-1412	22	10	of	of	ADP
ap-1412	22	11	complex	complex	ADJ
ap-1412	22	12	numbers	number	NOUN
ap-1412	22	13	)	)	PUNCT
ap-1412	22	14	such	such	ADJ
ap-1412	22	15	that	that	SCONJ
ap-1412	22	16	(	(	PUNCT
ap-1412	22	17	x	x	NOUN
ap-1412	22	18	,	,	PUNCT
ap-1412	22	19	αy+βz	αy+βz	NUM
ap-1412	22	20	)	)	PUNCT
ap-1412	22	21	=	=	SYM
ap-1412	22	22	α(x	α(x	PROPN
ap-1412	22	23	,	,	PUNCT
ap-1412	22	24	y	y	NOUN
ap-1412	22	25	)	)	PUNCT
ap-1412	22	26	+	+	CCONJ
ap-1412	22	27	β(x	β(x	PROPN
ap-1412	22	28	,	,	PUNCT
ap-1412	22	29	z	z	NOUN
ap-1412	22	30	)	)	PUNCT
ap-1412	22	31	for	for	ADP
ap-1412	22	32	all	all	DET
ap-1412	22	33	α	α	NOUN
ap-1412	22	34	,	,	PUNCT
ap-1412	22	35	β	β	X
ap-1412	22	36	∈	∈	PROPN
ap-1412	22	37	c	c	NOUN
ap-1412	22	38	and	and	CCONJ
ap-1412	22	39	x	x	PROPN
ap-1412	22	40	,	,	PUNCT
ap-1412	22	41	y	y	PROPN
ap-1412	22	42	,	,	PUNCT
ap-1412	22	43	z	z	PROPN
ap-1412	22	44	∈	∈	PROPN
ap-1412	22	45	h.	h.	PROPN
ap-1412	23	1	moreover	moreover	ADV
ap-1412	23	2	,	,	PUNCT
ap-1412	23	3	(	(	PUNCT
ap-1412	23	4	x	x	X
ap-1412	23	5	,	,	PUNCT
ap-1412	23	6	y	y	NOUN
ap-1412	23	7	)	)	PUNCT
ap-1412	24	1	=	=	SYM
ap-1412	24	2	(	(	PUNCT
ap-1412	24	3	y	y	PROPN
ap-1412	24	4	,	,	PUNCT
ap-1412	24	5	x	x	NOUN
ap-1412	24	6	)	)	PUNCT
ap-1412	24	7	and	and	CCONJ
ap-1412	24	8	finally	finally	ADV
ap-1412	24	9	(	(	PUNCT
ap-1412	24	10	x	x	NOUN
ap-1412	24	11	,	,	PUNCT
ap-1412	24	12	x	x	X
ap-1412	24	13	)	)	PUNCT
ap-1412	24	14	≥	≥	NOUN
ap-1412	24	15	0	0	NUM
ap-1412	24	16	at	at	ADP
ap-1412	24	17	which	which	PRON
ap-1412	24	18	(	(	PUNCT
ap-1412	24	19	x	x	NOUN
ap-1412	24	20	,	,	PUNCT
ap-1412	24	21	x	x	X
ap-1412	24	22	)	)	PUNCT
ap-1412	24	23	=	=	SYM
ap-1412	24	24	0	0	NUM
ap-1412	24	25	iff	iff	NOUN
ap-1412	24	26	x	x	PROPN
ap-1412	24	27	=	=	NOUN
ap-1412	24	28	0	0	NUM
ap-1412	24	29	.	.	PUNCT
ap-1412	25	1	the	the	DET
ap-1412	25	2	term	term	NOUN
ap-1412	25	3	dimension	dimension	NOUN
ap-1412	25	4	of	of	ADP
ap-1412	25	5	h	h	NOUN
ap-1412	25	6	in	in	ADP
ap-1412	25	7	the	the	DET
ap-1412	25	8	following	following	NOUN
ap-1412	25	9	always	always	ADV
ap-1412	25	10	means	mean	VERB
ap-1412	25	11	the	the	DET
ap-1412	25	12	hilbertian	hilbertian	ADJ
ap-1412	25	13	dimension	dimension	NOUN
ap-1412	25	14	defined	define	VERB
ap-1412	25	15	as	as	ADP
ap-1412	25	16	the	the	DET
ap-1412	25	17	cardinality	cardinality	NOUN
ap-1412	25	18	of	of	ADP
ap-1412	25	19	any	any	DET
ap-1412	25	20	orthonormal	orthonormal	ADJ
ap-1412	25	21	basis	basis	NOUN
ap-1412	25	22	of	of	ADP
ap-1412	25	23	h	h	PROPN
ap-1412	25	24	(	(	PUNCT
ap-1412	25	25	see	see	VERB
ap-1412	25	26	[	[	X
ap-1412	25	27	3	3	NUM
ap-1412	25	28	,	,	PUNCT
ap-1412	25	29	p.	p.	NOUN
ap-1412	25	30	44	44	NUM
ap-1412	25	31	]	]	SYM
ap-1412	25	32	)	)	PUNCT
ap-1412	25	33	.	.	PUNCT
ap-1412	26	1	moreover	moreover	ADV
ap-1412	26	2	,	,	PUNCT
ap-1412	26	3	we	we	PRON
ap-1412	26	4	will	will	AUX
ap-1412	26	5	assume	assume	VERB
ap-1412	26	6	that	that	SCONJ
ap-1412	26	7	all	all	PRON
ap-1412	26	8	considered	consider	VERB
ap-1412	26	9	linear	linear	PROPN
ap-1412	26	10	operators	operator	NOUN
ap-1412	26	11	a	a	DET
ap-1412	26	12	(	(	PUNCT
ap-1412	26	13	i.e.	i.e.	X
ap-1412	26	14	,	,	PUNCT
ap-1412	26	15	linear	linear	PROPN
ap-1412	26	16	maps	map	VERB
ap-1412	26	17	a	a	DET
ap-1412	26	18	:	:	PUNCT
ap-1412	26	19	d(a	d(a	PROPN
ap-1412	26	20	)	)	PUNCT
ap-1412	26	21	→	→	SYM
ap-1412	26	22	h	h	X
ap-1412	26	23	)	)	PUNCT
ap-1412	26	24	have	have	VERB
ap-1412	26	25	a	a	DET
ap-1412	26	26	domain	domain	NOUN
ap-1412	26	27	d(a	d(a	PROPN
ap-1412	26	28	)	)	PUNCT
ap-1412	26	29	a	a	DET
ap-1412	26	30	linear	linear	ADJ
ap-1412	26	31	subspace	subspace	NOUN
ap-1412	26	32	dense	dense	ADJ
ap-1412	26	33	in	in	ADP
ap-1412	26	34	h	h	NOUN
ap-1412	26	35	with	with	ADP
ap-1412	26	36	respect	respect	NOUN
ap-1412	26	37	to	to	ADP
ap-1412	26	38	metric	metric	ADJ
ap-1412	26	39	topology	topology	NOUN
ap-1412	26	40	induced	induce	VERB
ap-1412	26	41	by	by	ADP
ap-1412	26	42	inner	inner	ADJ
ap-1412	26	43	product	product	NOUN
ap-1412	26	44	,	,	PUNCT
ap-1412	26	45	so	so	ADV
ap-1412	26	46	d(a	d(a	PROPN
ap-1412	26	47	)	)	PUNCT
ap-1412	27	1	=	=	SYM
ap-1412	27	2	h.	h.	PROPN
ap-1412	28	1	moreover	moreover	ADV
ap-1412	28	2	,	,	PUNCT
ap-1412	28	3	our	our	PRON
ap-1412	28	4	next	next	ADJ
ap-1412	28	5	results	result	NOUN
ap-1412	28	6	will	will	AUX
ap-1412	28	7	be	be	AUX
ap-1412	28	8	for	for	ADP
ap-1412	28	9	positive	positive	ADJ
ap-1412	28	10	linear	linear	PROPN
ap-1412	28	11	operators	operator	NOUN
ap-1412	28	12	a	a	PRON
ap-1412	28	13	(	(	PUNCT
ap-1412	28	14	denoted	denote	VERB
ap-1412	28	15	by	by	ADP
ap-1412	28	16	a	a	DET
ap-1412	28	17	≥	≥	NOUN
ap-1412	28	18	0	0	NUM
ap-1412	28	19	)	)	PUNCT
ap-1412	28	20	,	,	PUNCT
ap-1412	28	21	meaning	mean	VERB
ap-1412	28	22	that	that	SCONJ
ap-1412	28	23	(	(	PUNCT
ap-1412	28	24	ax	ax	NOUN
ap-1412	28	25	,	,	PUNCT
ap-1412	28	26	x	x	NOUN
ap-1412	28	27	)	)	PUNCT
ap-1412	28	28	≥	≥	X
ap-1412	28	29	0	0	NUM
ap-1412	28	30	for	for	ADP
ap-1412	28	31	all	all	DET
ap-1412	28	32	x	x	SYM
ap-1412	28	33	∈	∈	PROPN
ap-1412	28	34	d(a	d(a	PROPN
ap-1412	28	35	)	)	PUNCT
ap-1412	28	36	,	,	PUNCT
ap-1412	28	37	therefore	therefore	ADV
ap-1412	28	38	operators	operator	VERB
ap-1412	28	39	a	a	PRON
ap-1412	28	40	are	be	AUX
ap-1412	28	41	also	also	ADV
ap-1412	28	42	symmetric	symmetric	ADJ
ap-1412	28	43	,	,	PUNCT
ap-1412	28	44	(	(	PUNCT
ap-1412	28	45	for	for	ADP
ap-1412	28	46	more	more	ADJ
ap-1412	28	47	details	detail	NOUN
ap-1412	28	48	see	see	VERB
ap-1412	28	49	[	[	X
ap-1412	28	50	3	3	NUM
ap-1412	28	51	]	]	NUM
ap-1412	28	52	)	)	PUNCT
ap-1412	28	53	.	.	PUNCT
ap-1412	29	1	we	we	PRON
ap-1412	29	2	will	will	AUX
ap-1412	29	3	denote	denote	VERB
ap-1412	29	4	the	the	DET
ap-1412	29	5	set	set	NOUN
ap-1412	29	6	of	of	ADP
ap-1412	29	7	all	all	DET
ap-1412	29	8	such	such	ADJ
ap-1412	29	9	operators	operator	NOUN
ap-1412	29	10	by	by	ADP
ap-1412	29	11	v(h	v(h	NOUN
ap-1412	29	12	)	)	PUNCT
ap-1412	29	13	.	.	PUNCT
ap-1412	30	1	recall	recall	VERB
ap-1412	30	2	that	that	PRON
ap-1412	30	3	a	a	DET
ap-1412	30	4	:	:	PUNCT
ap-1412	30	5	d(a	d(a	PROPN
ap-1412	30	6	)	)	PUNCT
ap-1412	30	7	→	→	SYM
ap-1412	30	8	h	h	NOUN
ap-1412	30	9	is	be	AUX
ap-1412	30	10	called	call	VERB
ap-1412	30	11	a	a	DET
ap-1412	30	12	bounded	bounded	ADJ
ap-1412	30	13	operator	operator	NOUN
ap-1412	30	14	if	if	SCONJ
ap-1412	30	15	there	there	PRON
ap-1412	30	16	exists	exist	VERB
ap-1412	30	17	a	a	DET
ap-1412	30	18	real	real	ADJ
ap-1412	30	19	constant	constant	ADJ
ap-1412	30	20	c	c	PROPN
ap-1412	30	21	≥	≥	NOUN
ap-1412	30	22	0	0	NUM
ap-1412	30	23	such	such	ADJ
ap-1412	30	24	that	that	SCONJ
ap-1412	30	25	‖ax‖	‖ax‖	ADJ
ap-1412	30	26	≤	≤	NOUN
ap-1412	30	27	c‖x‖	c‖x‖	NOUN
ap-1412	30	28	for	for	ADP
ap-1412	30	29	all	all	DET
ap-1412	30	30	x	x	SYM
ap-1412	30	31	∈	∈	PROPN
ap-1412	30	32	d(a	d(a	PROPN
ap-1412	30	33	)	)	PUNCT
ap-1412	30	34	and	and	CCONJ
ap-1412	30	35	hence	hence	ADV
ap-1412	30	36	a	a	PRON
ap-1412	30	37	is	be	AUX
ap-1412	30	38	an	an	DET
ap-1412	30	39	unbounded	unbounded	ADJ
ap-1412	30	40	operator	operator	NOUN
ap-1412	30	41	if	if	SCONJ
ap-1412	30	42	to	to	ADP
ap-1412	30	43	every	every	DET
ap-1412	30	44	c	c	NOUN
ap-1412	30	45	there	there	PRON
ap-1412	30	46	exists	exist	VERB
ap-1412	30	47	xc	xc	PROPN
ap-1412	30	48	∈	∈	PROPN
ap-1412	30	49	d(a	d(a	PROPN
ap-1412	30	50	)	)	PUNCT
ap-1412	30	51	with	with	ADP
ap-1412	30	52	‖axc‖	‖axc‖	ADV
ap-1412	30	53	>	>	X
ap-1412	30	54	c‖xc‖.	c‖xc‖.	NOUN
ap-1412	30	55	to	to	ADP
ap-1412	30	56	every	every	DET
ap-1412	30	57	linear	linear	ADJ
ap-1412	30	58	operator	operator	NOUN
ap-1412	30	59	with	with	ADP
ap-1412	30	60	d(a	d(a	PROPN
ap-1412	30	61	)	)	PUNCT
ap-1412	31	1	=	=	NOUN
ap-1412	32	1	h	h	NOUN
ap-1412	32	2	there	there	PRON
ap-1412	32	3	exists	exist	VERB
ap-1412	32	4	the	the	DET
ap-1412	32	5	adjoint	adjoint	PROPN
ap-1412	32	6	linear	linear	PROPN
ap-1412	32	7	operator	operator	NOUN
ap-1412	32	8	a∗	a∗	NOUN
ap-1412	32	9	of	of	ADP
ap-1412	32	10	a	a	DET
ap-1412	32	11	such	such	ADJ
ap-1412	32	12	that	that	DET
ap-1412	32	13	d(a∗	d(a∗	NOUN
ap-1412	32	14	)	)	PUNCT
ap-1412	32	15	=	=	SYM
ap-1412	33	1	{	{	PUNCT
ap-1412	33	2	y	y	PROPN
ap-1412	33	3	∈	∈	PROPN
ap-1412	33	4	h	h	NOUN
ap-1412	33	5	|	|	ADV
ap-1412	33	6	there	there	PRON
ap-1412	33	7	exists	exist	VERB
ap-1412	33	8	y∗	y∗	PROPN
ap-1412	33	9	∈	∈	PROPN
ap-1412	33	10	h	h	NOUN
ap-1412	33	11	such	such	ADJ
ap-1412	33	12	that	that	PRON
ap-1412	33	13	(	(	PUNCT
ap-1412	33	14	y∗	y∗	ADV
ap-1412	33	15	,	,	PUNCT
ap-1412	33	16	x	x	NOUN
ap-1412	33	17	)	)	PUNCT
ap-1412	33	18	=	=	SYM
ap-1412	34	1	(	(	PUNCT
ap-1412	34	2	y	y	NOUN
ap-1412	34	3	,	,	PUNCT
ap-1412	34	4	ax	ax	NOUN
ap-1412	34	5	)	)	PUNCT
ap-1412	34	6	for	for	ADP
ap-1412	34	7	all	all	DET
ap-1412	34	8	x	x	PROPN
ap-1412	34	9	∈	∈	PROPN
ap-1412	34	10	d(a	d(a	PROPN
ap-1412	34	11	)	)	PUNCT
ap-1412	34	12	}	}	PUNCT
ap-1412	34	13	and	and	CCONJ
ap-1412	34	14	a∗y	a∗y	NUM
ap-1412	34	15	=	=	PUNCT
ap-1412	34	16	y∗	y∗	ADV
ap-1412	34	17	for	for	ADP
ap-1412	34	18	every	every	DET
ap-1412	34	19	y	y	PROPN
ap-1412	34	20	∈	∈	PROPN
ap-1412	34	21	d(a∗	d(a∗	NUM
ap-1412	34	22	)	)	PUNCT
ap-1412	34	23	.	.	PUNCT
ap-1412	35	1	78	78	NUM
ap-1412	35	2	acta	acta	PROPN
ap-1412	35	3	polytechnica	polytechnica	PROPN
ap-1412	35	4	vol	vol	NOUN
ap-1412	35	5	.	.	PUNCT
ap-1412	36	1	51	51	NUM
ap-1412	36	2	no	no	INTJ
ap-1412	36	3	.	.	PUNCT
ap-1412	36	4	4/2011	4/2011	NUM
ap-1412	37	1	if	if	SCONJ
ap-1412	37	2	a∗	a∗	NOUN
ap-1412	37	3	=	=	PUNCT
ap-1412	37	4	a	a	DET
ap-1412	37	5	then	then	ADV
ap-1412	37	6	a	a	PRON
ap-1412	37	7	is	be	AUX
ap-1412	37	8	called	call	VERB
ap-1412	37	9	self	self	NOUN
ap-1412	37	10	-	-	PUNCT
ap-1412	37	11	adjoint	adjoint	NOUN
ap-1412	37	12	.	.	PUNCT
ap-1412	38	1	the	the	DET
ap-1412	38	2	set	set	NOUN
ap-1412	38	3	of	of	ADP
ap-1412	38	4	all	all	DET
ap-1412	38	5	positive	positive	ADJ
ap-1412	38	6	self	self	NOUN
ap-1412	38	7	-	-	PUNCT
ap-1412	38	8	adjoint	adjoint	NOUN
ap-1412	38	9	linear	linear	PROPN
ap-1412	38	10	operators	operator	NOUN
ap-1412	38	11	densely	densely	ADV
ap-1412	38	12	defined	define	VERB
ap-1412	38	13	in	in	ADP
ap-1412	38	14	h	h	NOUN
ap-1412	38	15	will	will	AUX
ap-1412	38	16	be	be	AUX
ap-1412	38	17	denoted	denote	VERB
ap-1412	38	18	by	by	ADP
ap-1412	38	19	sp(h	sp(h	NOUN
ap-1412	38	20	)	)	PUNCT
ap-1412	38	21	.	.	PUNCT
ap-1412	39	1	hence	hence	ADV
ap-1412	39	2	sp(h	sp(h	VERB
ap-1412	39	3	)	)	PUNCT
ap-1412	40	1	=	=	SYM
ap-1412	40	2	{	{	PUNCT
ap-1412	40	3	a	a	DET
ap-1412	40	4	∈	∈	PROPN
ap-1412	40	5	v(h	v(h	NOUN
ap-1412	40	6	)	)	PUNCT
ap-1412	40	7	|	|	ADV
ap-1412	40	8	a	a	DET
ap-1412	40	9	=	=	NOUN
ap-1412	40	10	a∗	a∗	NOUN
ap-1412	40	11	}	}	PUNCT
ap-1412	40	12	.	.	PUNCT
ap-1412	41	1	a	a	DET
ap-1412	41	2	densely	densely	ADV
ap-1412	41	3	defined	define	VERB
ap-1412	41	4	linear	linear	NOUN
ap-1412	41	5	operator	operator	NOUN
ap-1412	41	6	a	a	PRON
ap-1412	41	7	on	on	ADP
ap-1412	41	8	h	h	NOUN
ap-1412	41	9	is	be	AUX
ap-1412	41	10	called	call	VERB
ap-1412	41	11	symmetric	symmetric	ADJ
ap-1412	41	12	,	,	PUNCT
ap-1412	41	13	if	if	SCONJ
ap-1412	41	14	a	a	DET
ap-1412	41	15	⊂	⊂	PROPN
ap-1412	41	16	a∗.	a∗.	NOUN
ap-1412	41	17	here	here	ADV
ap-1412	41	18	we	we	PRON
ap-1412	41	19	write	write	VERB
ap-1412	41	20	a	a	DET
ap-1412	41	21	⊂	⊂	PROPN
ap-1412	41	22	b	b	PROPN
ap-1412	41	23	iff	iff	PROPN
ap-1412	41	24	d(a	d(a	PROPN
ap-1412	41	25	)	)	PUNCT
ap-1412	41	26	⊆	⊆	NUM
ap-1412	41	27	d(b	d(b	NOUN
ap-1412	41	28	)	)	PUNCT
ap-1412	41	29	and	and	CCONJ
ap-1412	41	30	ax	ax	NOUN
ap-1412	41	31	=	=	NOUN
ap-1412	41	32	bx	bx	PROPN
ap-1412	41	33	for	for	ADP
ap-1412	41	34	every	every	DET
ap-1412	41	35	x	x	PROPN
ap-1412	41	36	∈	∈	PROPN
ap-1412	41	37	d(a	d(a	PROPN
ap-1412	41	38	)	)	PUNCT
ap-1412	41	39	.	.	PUNCT
ap-1412	42	1	the	the	DET
ap-1412	42	2	condition	condition	NOUN
ap-1412	42	3	a	a	DET
ap-1412	42	4	⊂	⊂	PROPN
ap-1412	42	5	a∗	a∗	PROPN
ap-1412	42	6	is	be	AUX
ap-1412	42	7	equivalent	equivalent	ADJ
ap-1412	42	8	to	to	ADP
ap-1412	42	9	(	(	PUNCT
ap-1412	42	10	y	y	NOUN
ap-1412	42	11	,	,	PUNCT
ap-1412	42	12	ax	ax	NOUN
ap-1412	42	13	)	)	PUNCT
ap-1412	42	14	=	=	SYM
ap-1412	42	15	(	(	PUNCT
ap-1412	42	16	ay	ay	INTJ
ap-1412	42	17	,	,	PUNCT
ap-1412	42	18	x	x	NOUN
ap-1412	42	19	)	)	PUNCT
ap-1412	42	20	for	for	ADP
ap-1412	42	21	all	all	DET
ap-1412	42	22	x	x	NOUN
ap-1412	42	23	,	,	PUNCT
ap-1412	42	24	y	y	PROPN
ap-1412	42	25	∈	∈	PROPN
ap-1412	42	26	d(a	d(a	PROPN
ap-1412	42	27	)	)	PUNCT
ap-1412	42	28	.	.	PUNCT
ap-1412	43	1	an	an	DET
ap-1412	43	2	operator	operator	NOUN
ap-1412	43	3	a	a	PRON
ap-1412	43	4	:	:	PUNCT
ap-1412	43	5	d(a	d(a	PROPN
ap-1412	43	6	)	)	PUNCT
ap-1412	43	7	→	→	SYM
ap-1412	43	8	h	h	NOUN
ap-1412	43	9	is	be	AUX
ap-1412	43	10	called	call	VERB
ap-1412	43	11	closed	closed	ADJ
ap-1412	43	12	if	if	SCONJ
ap-1412	43	13	for	for	ADP
ap-1412	43	14	every	every	DET
ap-1412	43	15	sequence	sequence	NOUN
ap-1412	43	16	(	(	PUNCT
ap-1412	43	17	xn)n∈n	xn)n∈n	NUM
ap-1412	43	18	,	,	PUNCT
ap-1412	43	19	xn	xn	PROPN
ap-1412	43	20	∈	∈	PROPN
ap-1412	43	21	d(a	d(a	PROPN
ap-1412	43	22	)	)	PUNCT
ap-1412	43	23	,	,	PUNCT
ap-1412	43	24	such	such	ADJ
ap-1412	43	25	that	that	PRON
ap-1412	43	26	xn	xn	PUNCT
ap-1412	44	1	→	→	SYM
ap-1412	44	2	x	x	SYM
ap-1412	44	3	∈	∈	PROPN
ap-1412	44	4	h	h	NOUN
ap-1412	44	5	and	and	CCONJ
ap-1412	44	6	axn	axn	PROPN
ap-1412	44	7	→	→	SYM
ap-1412	44	8	y	y	PROPN
ap-1412	44	9	∈	∈	PROPN
ap-1412	44	10	h	h	NOUN
ap-1412	44	11	as	as	ADP
ap-1412	44	12	n	n	PROPN
ap-1412	44	13	→	→	SYM
ap-1412	44	14	∞	∞	NUM
ap-1412	44	15	one	one	NOUN
ap-1412	44	16	has	have	VERB
ap-1412	44	17	x	x	PROPN
ap-1412	44	18	∈	∈	PROPN
ap-1412	44	19	d(a	d(a	PROPN
ap-1412	44	20	)	)	PUNCT
ap-1412	44	21	and	and	CCONJ
ap-1412	44	22	ax	ax	NOUN
ap-1412	44	23	=	=	PUNCT
ap-1412	44	24	y.	y.	NOUN
ap-1412	44	25	since	since	SCONJ
ap-1412	44	26	a	a	DET
ap-1412	44	27	∈	∈	PROPN
ap-1412	44	28	v(h	v(h	NOUN
ap-1412	44	29	)	)	PUNCT
ap-1412	44	30	is	be	AUX
ap-1412	44	31	positive	positive	ADJ
ap-1412	44	32	and	and	CCONJ
ap-1412	44	33	hence	hence	ADV
ap-1412	44	34	also	also	ADV
ap-1412	44	35	symmetric	symmetric	ADJ
ap-1412	44	36	(	(	PUNCT
ap-1412	44	37	see	see	VERB
ap-1412	44	38	[	[	X
ap-1412	44	39	3	3	NUM
ap-1412	44	40	]	]	PUNCT
ap-1412	44	41	,	,	PUNCT
ap-1412	44	42	p.	p.	NOUN
ap-1412	44	43	142	142	NUM
ap-1412	44	44	)	)	PUNCT
ap-1412	44	45	there	there	PRON
ap-1412	44	46	exists	exist	VERB
ap-1412	44	47	a	a	DET
ap-1412	44	48	closed	closed	ADJ
ap-1412	44	49	operator	operator	NOUN
ap-1412	44	50	a	a	DET
ap-1412	44	51	such	such	ADJ
ap-1412	44	52	that	that	SCONJ
ap-1412	44	53	a	a	DET
ap-1412	44	54	⊂	⊂	PROPN
ap-1412	44	55	a	a	PROPN
ap-1412	44	56	and	and	CCONJ
ap-1412	44	57	a	a	DET
ap-1412	44	58	⊂	⊂	PROPN
ap-1412	44	59	b	b	PROPN
ap-1412	44	60	for	for	ADP
ap-1412	44	61	every	every	DET
ap-1412	44	62	closed	closed	ADJ
ap-1412	44	63	operator	operator	NOUN
ap-1412	44	64	b	b	PROPN
ap-1412	44	65	⊃	⊃	PROPN
ap-1412	44	66	a.	a.	NOUN
ap-1412	44	67	moreover	moreover	ADV
ap-1412	44	68	a	a	PRON
ap-1412	44	69	is	be	AUX
ap-1412	44	70	symmetric	symmetric	ADJ
ap-1412	44	71	and	and	CCONJ
ap-1412	44	72	it	it	PRON
ap-1412	44	73	is	be	AUX
ap-1412	44	74	called	call	VERB
ap-1412	44	75	the	the	DET
ap-1412	44	76	closure	closure	NOUN
ap-1412	44	77	of	of	ADP
ap-1412	44	78	a.	a.	NOUN
ap-1412	44	79	a	a	DET
ap-1412	44	80	symmetric	symmetric	ADJ
ap-1412	44	81	operator	operator	NOUN
ap-1412	44	82	is	be	AUX
ap-1412	44	83	called	call	VERB
ap-1412	44	84	essentially	essentially	ADV
ap-1412	44	85	self	self	NOUN
ap-1412	44	86	-	-	PUNCT
ap-1412	44	87	adjoint	adjoint	NOUN
ap-1412	44	88	if	if	SCONJ
ap-1412	44	89	(	(	PUNCT
ap-1412	44	90	a)∗	a)∗	PROPN
ap-1412	44	91	=	=	PUNCT
ap-1412	44	92	a	a	PROPN
ap-1412	45	1	and	and	CCONJ
ap-1412	45	2	then	then	ADV
ap-1412	45	3	a	a	PRON
ap-1412	45	4	is	be	AUX
ap-1412	45	5	a	a	DET
ap-1412	45	6	unique	unique	ADJ
ap-1412	45	7	self	self	NOUN
ap-1412	45	8	-	-	PUNCT
ap-1412	45	9	adjoint	adjoint	NOUN
ap-1412	45	10	extension	extension	NOUN
ap-1412	45	11	of	of	ADP
ap-1412	45	12	a	a	DET
ap-1412	45	13	[	[	X
ap-1412	45	14	3	3	NUM
ap-1412	45	15	,	,	PUNCT
ap-1412	45	16	p.	p.	NOUN
ap-1412	45	17	96	96	NUM
ap-1412	45	18	]	]	PUNCT
ap-1412	45	19	.	.	PUNCT
ap-1412	46	1	we	we	PRON
ap-1412	46	2	shall	shall	AUX
ap-1412	46	3	show	show	VERB
ap-1412	46	4	in	in	ADP
ap-1412	46	5	section	section	NOUN
ap-1412	46	6	3	3	NUM
ap-1412	46	7	that	that	PRON
ap-1412	46	8	,	,	PUNCT
ap-1412	46	9	under	under	ADP
ap-1412	46	10	a	a	DET
ap-1412	46	11	partially	partially	ADV
ap-1412	46	12	defined	define	VERB
ap-1412	46	13	usual	usual	ADJ
ap-1412	46	14	sum	sum	NOUN
ap-1412	46	15	of	of	ADP
ap-1412	46	16	linear	linear	PROPN
ap-1412	46	17	operators	operator	NOUN
ap-1412	46	18	,	,	PUNCT
ap-1412	46	19	sets	set	VERB
ap-1412	46	20	v(h	v(h	NOUN
ap-1412	46	21	)	)	PUNCT
ap-1412	46	22	and	and	CCONJ
ap-1412	46	23	sp(h	sp(h	NOUN
ap-1412	46	24	)	)	PUNCT
ap-1412	46	25	form	form	VERB
ap-1412	46	26	quantum	quantum	NOUN
ap-1412	46	27	structures	structure	NOUN
ap-1412	46	28	called	call	VERB
ap-1412	46	29	(	(	PUNCT
ap-1412	46	30	generalized	generalized	ADJ
ap-1412	46	31	)	)	PUNCT
ap-1412	46	32	effect	effect	NOUN
ap-1412	46	33	algebras	algebra	NOUN
ap-1412	46	34	(	(	PUNCT
ap-1412	46	35	see	see	VERB
ap-1412	46	36	also	also	ADV
ap-1412	46	37	[	[	X
ap-1412	46	38	9	9	NUM
ap-1412	46	39	]	]	PUNCT
ap-1412	46	40	)	)	PUNCT
ap-1412	46	41	.	.	PUNCT
ap-1412	47	1	now	now	ADV
ap-1412	47	2	we	we	PRON
ap-1412	47	3	recall	recall	VERB
ap-1412	47	4	their	their	PRON
ap-1412	47	5	definitions	definition	NOUN
ap-1412	47	6	.	.	PUNCT
ap-1412	48	1	definition	definition	NOUN
ap-1412	48	2	2.1	2.1	NUM
ap-1412	48	3	(	(	PUNCT
ap-1412	48	4	foulis	foulis	PROPN
ap-1412	48	5	and	and	CCONJ
ap-1412	48	6	bennett	bennett	PROPN
ap-1412	48	7	,	,	PUNCT
ap-1412	48	8	1994	1994	NUM
ap-1412	48	9	)	)	PUNCT
ap-1412	48	10	a	a	DET
ap-1412	48	11	partial	partial	ADJ
ap-1412	48	12	algebra	algebra	NOUN
ap-1412	48	13	(	(	PUNCT
ap-1412	48	14	e;⊕	e;⊕	ADJ
ap-1412	48	15	,	,	PUNCT
ap-1412	48	16	0	0	NUM
ap-1412	48	17	,	,	PUNCT
ap-1412	48	18	1	1	NUM
ap-1412	48	19	)	)	PUNCT
ap-1412	48	20	is	be	AUX
ap-1412	48	21	called	call	VERB
ap-1412	48	22	an	an	DET
ap-1412	48	23	effect	effect	NOUN
ap-1412	48	24	algebra	algebra	NOUN
ap-1412	48	25	if	if	SCONJ
ap-1412	48	26	0,1	0,1	NUM
ap-1412	48	27	are	be	AUX
ap-1412	48	28	two	two	NUM
ap-1412	48	29	distinguished	distinguished	ADJ
ap-1412	48	30	elements	element	NOUN
ap-1412	48	31	and	and	CCONJ
ap-1412	48	32	⊕	⊕	PROPN
ap-1412	48	33	is	be	AUX
ap-1412	48	34	a	a	DET
ap-1412	48	35	partially	partially	ADV
ap-1412	48	36	defined	define	VERB
ap-1412	48	37	binary	binary	ADJ
ap-1412	48	38	operation	operation	NOUN
ap-1412	48	39	on	on	ADP
ap-1412	48	40	e	e	PROPN
ap-1412	48	41	which	which	PRON
ap-1412	48	42	satisfies	satisfy	VERB
ap-1412	48	43	the	the	DET
ap-1412	48	44	following	follow	VERB
ap-1412	48	45	conditions	condition	NOUN
ap-1412	48	46	for	for	ADP
ap-1412	48	47	any	any	DET
ap-1412	48	48	x	x	NOUN
ap-1412	48	49	,	,	PUNCT
ap-1412	48	50	y	y	PROPN
ap-1412	48	51	,	,	PUNCT
ap-1412	48	52	z	z	NOUN
ap-1412	48	53	∈	∈	PROPN
ap-1412	49	1	e	e	NOUN
ap-1412	49	2	:	:	PUNCT
ap-1412	49	3	(	(	PUNCT
ap-1412	49	4	e1	e1	NOUN
ap-1412	49	5	)	)	PUNCT
ap-1412	49	6	x	x	PUNCT
ap-1412	49	7	⊕	⊕	NOUN
ap-1412	49	8	y	y	NOUN
ap-1412	49	9	=	=	SYM
ap-1412	49	10	y	y	PROPN
ap-1412	49	11	⊕	⊕	PROPN
ap-1412	49	12	x	x	PUNCT
ap-1412	50	1	if	if	SCONJ
ap-1412	50	2	x	x	PROPN
ap-1412	50	3	⊕	⊕	NOUN
ap-1412	50	4	y	y	PROPN
ap-1412	50	5	is	be	AUX
ap-1412	50	6	defined	define	VERB
ap-1412	50	7	,	,	PUNCT
ap-1412	50	8	(	(	PUNCT
ap-1412	50	9	e2	e2	PROPN
ap-1412	50	10	)	)	PUNCT
ap-1412	50	11	(	(	PUNCT
ap-1412	50	12	x⊕y)⊕z	x⊕y)⊕z	X
ap-1412	50	13	=	=	SYM
ap-1412	50	14	x⊕(y⊕z	x⊕(y⊕z	NOUN
ap-1412	50	15	)	)	PUNCT
ap-1412	50	16	if	if	SCONJ
ap-1412	50	17	one	one	NUM
ap-1412	50	18	side	side	NOUN
ap-1412	50	19	is	be	AUX
ap-1412	50	20	defined	define	VERB
ap-1412	50	21	,	,	PUNCT
ap-1412	50	22	(	(	PUNCT
ap-1412	50	23	e3	e3	NOUN
ap-1412	50	24	)	)	PUNCT
ap-1412	50	25	for	for	ADP
ap-1412	50	26	every	every	DET
ap-1412	50	27	x	x	SYM
ap-1412	50	28	∈	∈	PROPN
ap-1412	50	29	e	e	NOUN
ap-1412	50	30	there	there	PRON
ap-1412	50	31	exists	exist	VERB
ap-1412	50	32	a	a	DET
ap-1412	50	33	unique	unique	ADJ
ap-1412	50	34	y	y	PROPN
ap-1412	50	35	∈	∈	PROPN
ap-1412	50	36	e	e	NOUN
ap-1412	50	37	such	such	ADJ
ap-1412	50	38	that	that	SCONJ
ap-1412	50	39	x	x	PROPN
ap-1412	50	40	⊕	⊕	NOUN
ap-1412	50	41	y	y	NOUN
ap-1412	50	42	=	=	SYM
ap-1412	50	43	1	1	NUM
ap-1412	50	44	(	(	PUNCT
ap-1412	50	45	we	we	PRON
ap-1412	50	46	put	put	VERB
ap-1412	50	47	x′	x′	PROPN
ap-1412	50	48	=	=	SYM
ap-1412	50	49	y	y	PROPN
ap-1412	50	50	)	)	PUNCT
ap-1412	50	51	,	,	PUNCT
ap-1412	50	52	(	(	PUNCT
ap-1412	50	53	e4	e4	PROPN
ap-1412	50	54	)	)	PUNCT
ap-1412	50	55	if	if	SCONJ
ap-1412	50	56	1	1	NUM
ap-1412	50	57	⊕	⊕	PROPN
ap-1412	50	58	x	x	PUNCT
ap-1412	50	59	is	be	AUX
ap-1412	50	60	defined	define	VERB
ap-1412	50	61	then	then	ADV
ap-1412	50	62	x	x	X
ap-1412	50	63	=	=	NOUN
ap-1412	51	1	0	0	X
ap-1412	51	2	.	.	PUNCT
ap-1412	52	1	we	we	PRON
ap-1412	52	2	often	often	ADV
ap-1412	52	3	denote	denote	VERB
ap-1412	52	4	the	the	DET
ap-1412	52	5	effect	effect	NOUN
ap-1412	52	6	algebra	algebra	NOUN
ap-1412	52	7	(	(	PUNCT
ap-1412	52	8	e;⊕	e;⊕	ADJ
ap-1412	52	9	,	,	PUNCT
ap-1412	52	10	0	0	NUM
ap-1412	52	11	,	,	PUNCT
ap-1412	52	12	1	1	NUM
ap-1412	52	13	)	)	PUNCT
ap-1412	52	14	briefly	briefly	ADV
ap-1412	52	15	by	by	ADP
ap-1412	52	16	e.	e.	PROPN
ap-1412	52	17	on	on	ADP
ap-1412	52	18	every	every	DET
ap-1412	52	19	effect	effect	NOUN
ap-1412	52	20	algebra	algebra	NOUN
ap-1412	52	21	e	e	NOUN
ap-1412	52	22	the	the	DET
ap-1412	52	23	partial	partial	ADJ
ap-1412	52	24	order	order	NOUN
ap-1412	52	25	≤	≤	NOUN
ap-1412	52	26	and	and	CCONJ
ap-1412	52	27	partial	partial	ADJ
ap-1412	52	28	binary	binary	ADJ
ap-1412	52	29	operation	operation	NOUN
ap-1412	52	30	�	�	PROPN
ap-1412	52	31	can	can	AUX
ap-1412	52	32	be	be	AUX
ap-1412	52	33	introduced	introduce	VERB
ap-1412	52	34	as	as	SCONJ
ap-1412	52	35	follows	follow	VERB
ap-1412	52	36	:	:	PUNCT
ap-1412	52	37	x	x	SYM
ap-1412	52	38	≤	≤	X
ap-1412	52	39	y	y	PROPN
ap-1412	52	40	and	and	CCONJ
ap-1412	52	41	y	y	PROPN
ap-1412	52	42	�	�	PROPN
ap-1412	52	43	x	x	SYM
ap-1412	52	44	=	=	SYM
ap-1412	52	45	z	z	SYM
ap-1412	52	46	iff	iff	PROPN
ap-1412	52	47	x⊕z	x⊕z	PROPN
ap-1412	52	48	is	be	AUX
ap-1412	52	49	defined	define	VERB
ap-1412	52	50	and	and	CCONJ
ap-1412	52	51	x⊕z	x⊕z	PROPN
ap-1412	52	52	=	=	PUNCT
ap-1412	52	53	y.	y.	NOUN
ap-1412	52	54	if	if	SCONJ
ap-1412	52	55	e	e	NOUN
ap-1412	52	56	with	with	ADP
ap-1412	52	57	the	the	DET
ap-1412	52	58	defined	define	VERB
ap-1412	52	59	partial	partial	ADJ
ap-1412	52	60	order	order	NOUN
ap-1412	52	61	is	be	AUX
ap-1412	52	62	a	a	DET
ap-1412	52	63	lattice	lattice	NOUN
ap-1412	52	64	(	(	PUNCT
ap-1412	52	65	a	a	DET
ap-1412	52	66	complete	complete	ADJ
ap-1412	52	67	lattice	lattice	NOUN
ap-1412	52	68	)	)	PUNCT
ap-1412	52	69	then	then	ADV
ap-1412	52	70	(	(	PUNCT
ap-1412	52	71	e;⊕	e;⊕	ADJ
ap-1412	52	72	,	,	PUNCT
ap-1412	52	73	0	0	NUM
ap-1412	52	74	,	,	PUNCT
ap-1412	52	75	1	1	NUM
ap-1412	52	76	)	)	PUNCT
ap-1412	52	77	is	be	AUX
ap-1412	52	78	called	call	VERB
ap-1412	52	79	a	a	DET
ap-1412	52	80	lattice	lattice	ADJ
ap-1412	52	81	effect	effect	NOUN
ap-1412	52	82	algebra	algebra	NOUN
ap-1412	52	83	(	(	PUNCT
ap-1412	52	84	a	a	DET
ap-1412	52	85	complete	complete	ADJ
ap-1412	52	86	lattice	lattice	NOUN
ap-1412	52	87	effect	effect	NOUN
ap-1412	52	88	algebra	algebra	PROPN
ap-1412	52	89	)	)	PUNCT
ap-1412	52	90	.	.	PUNCT
ap-1412	53	1	generalizations	generalization	NOUN
ap-1412	53	2	of	of	ADP
ap-1412	53	3	effect	effect	NOUN
ap-1412	53	4	algebras	algebra	NOUN
ap-1412	53	5	(	(	PUNCT
ap-1412	53	6	i.e.	i.e.	X
ap-1412	53	7	,	,	PUNCT
ap-1412	53	8	without	without	ADP
ap-1412	53	9	a	a	DET
ap-1412	53	10	top	top	ADJ
ap-1412	53	11	element	element	NOUN
ap-1412	53	12	1	1	NUM
ap-1412	53	13	)	)	PUNCT
ap-1412	53	14	have	have	AUX
ap-1412	53	15	been	be	AUX
ap-1412	53	16	studied	study	VERB
ap-1412	53	17	by	by	ADP
ap-1412	53	18	kôpka	kôpka	PROPN
ap-1412	53	19	and	and	CCONJ
ap-1412	53	20	chovanec	chovanec	PROPN
ap-1412	53	21	(	(	PUNCT
ap-1412	53	22	1994	1994	NUM
ap-1412	53	23	)	)	PUNCT
ap-1412	53	24	(	(	PUNCT
ap-1412	53	25	difference	difference	NOUN
ap-1412	53	26	posets	poset	NOUN
ap-1412	53	27	)	)	PUNCT
ap-1412	53	28	,	,	PUNCT
ap-1412	53	29	foulis	foulis	PROPN
ap-1412	53	30	and	and	CCONJ
ap-1412	53	31	bennett	bennett	PROPN
ap-1412	53	32	(	(	PUNCT
ap-1412	53	33	1994	1994	NUM
ap-1412	53	34	)	)	PUNCT
ap-1412	53	35	(	(	PUNCT
ap-1412	53	36	cones	cone	NOUN
ap-1412	53	37	)	)	PUNCT
ap-1412	53	38	,	,	PUNCT
ap-1412	53	39	kalmbach	kalmbach	NOUN
ap-1412	53	40	and	and	CCONJ
ap-1412	53	41	riečanová	riečanová	PROPN
ap-1412	53	42	(	(	PUNCT
ap-1412	53	43	1994	1994	NUM
ap-1412	53	44	)	)	PUNCT
ap-1412	53	45	(	(	PUNCT
ap-1412	53	46	abelian	abelian	PROPN
ap-1412	53	47	ri	ri	NOUN
ap-1412	53	48	-	-	PUNCT
ap-1412	53	49	posets	poset	NOUN
ap-1412	53	50	and	and	CCONJ
ap-1412	53	51	abelian	abelian	PROPN
ap-1412	53	52	ri	ri	PROPN
ap-1412	53	53	semigroups	semigroup	NOUN
ap-1412	53	54	)	)	PUNCT
ap-1412	53	55	and	and	CCONJ
ap-1412	53	56	hedĺıková	hedĺıková	PROPN
ap-1412	53	57	and	and	CCONJ
ap-1412	53	58	pulmannová	pulmannová	ADJ
ap-1412	53	59	(	(	PUNCT
ap-1412	53	60	1996	1996	NUM
ap-1412	53	61	)	)	PUNCT
ap-1412	53	62	(	(	PUNCT
ap-1412	53	63	generalized	generalize	VERB
ap-1412	53	64	dposets	dposet	NOUN
ap-1412	53	65	and	and	CCONJ
ap-1412	53	66	cancellative	cancellative	ADJ
ap-1412	53	67	positive	positive	ADJ
ap-1412	53	68	partial	partial	ADJ
ap-1412	53	69	abelian	abelian	NOUN
ap-1412	53	70	semigroups	semigroup	NOUN
ap-1412	53	71	)	)	PUNCT
ap-1412	53	72	.	.	PUNCT
ap-1412	54	1	it	it	PRON
ap-1412	54	2	can	can	AUX
ap-1412	54	3	be	be	AUX
ap-1412	54	4	shown	show	VERB
ap-1412	54	5	that	that	SCONJ
ap-1412	54	6	all	all	DET
ap-1412	54	7	of	of	ADP
ap-1412	54	8	the	the	DET
ap-1412	54	9	above	above	ADJ
ap-1412	54	10	mentioned	mention	VERB
ap-1412	54	11	generalizations	generalization	NOUN
ap-1412	54	12	of	of	ADP
ap-1412	54	13	effect	effect	NOUN
ap-1412	54	14	algebras	algebra	NOUN
ap-1412	54	15	are	be	AUX
ap-1412	54	16	mutually	mutually	ADV
ap-1412	54	17	equivalent	equivalent	ADJ
ap-1412	54	18	and	and	CCONJ
ap-1412	54	19	extend	extend	VERB
ap-1412	54	20	similar	similar	ADJ
ap-1412	54	21	previous	previous	ADJ
ap-1412	54	22	results	result	NOUN
ap-1412	54	23	for	for	ADP
ap-1412	54	24	generalized	generalized	ADJ
ap-1412	54	25	boolean	boolean	ADJ
ap-1412	54	26	algebras	algebra	NOUN
ap-1412	54	27	and	and	CCONJ
ap-1412	54	28	orthomodular	orthomodular	ADJ
ap-1412	54	29	lattices	lattice	NOUN
ap-1412	54	30	and	and	CCONJ
ap-1412	54	31	posets	poset	NOUN
ap-1412	54	32	.	.	PUNCT
ap-1412	55	1	definition	definition	NOUN
ap-1412	55	2	2.2	2.2	NUM
ap-1412	55	3	(	(	PUNCT
ap-1412	55	4	1	1	NUM
ap-1412	55	5	)	)	PUNCT
ap-1412	55	6	a	a	DET
ap-1412	55	7	generalized	generalized	ADJ
ap-1412	55	8	effect	effect	NOUN
ap-1412	55	9	algebra	algebra	NOUN
ap-1412	55	10	(	(	PUNCT
ap-1412	55	11	e	e	NOUN
ap-1412	55	12	,	,	PUNCT
ap-1412	55	13	⊕	⊕	PROPN
ap-1412	55	14	,	,	PUNCT
ap-1412	55	15	0	0	NUM
ap-1412	55	16	)	)	PUNCT
ap-1412	55	17	is	be	AUX
ap-1412	55	18	a	a	DET
ap-1412	55	19	set	set	NOUN
ap-1412	55	20	e	e	NOUN
ap-1412	55	21	with	with	ADP
ap-1412	55	22	element	element	NOUN
ap-1412	55	23	0	0	NUM
ap-1412	55	24	∈	∈	PROPN
ap-1412	55	25	e	e	NOUN
ap-1412	55	26	and	and	CCONJ
ap-1412	55	27	partial	partial	ADJ
ap-1412	55	28	binary	binary	ADJ
ap-1412	55	29	operation	operation	NOUN
ap-1412	55	30	⊕	⊕	PROPN
ap-1412	55	31	satisfying	satisfy	VERB
ap-1412	55	32	for	for	ADP
ap-1412	55	33	any	any	DET
ap-1412	55	34	x	x	NOUN
ap-1412	55	35	,	,	PUNCT
ap-1412	55	36	y	y	PROPN
ap-1412	55	37	,	,	PUNCT
ap-1412	55	38	z	z	NOUN
ap-1412	55	39	∈	∈	PROPN
ap-1412	55	40	e	e	NOUN
ap-1412	55	41	conditions	condition	NOUN
ap-1412	55	42	(	(	PUNCT
ap-1412	55	43	ge1	ge1	NOUN
ap-1412	55	44	)	)	PUNCT
ap-1412	55	45	x	x	PUNCT
ap-1412	55	46	⊕	⊕	NOUN
ap-1412	55	47	y	y	NOUN
ap-1412	55	48	=	=	SYM
ap-1412	55	49	y	y	PROPN
ap-1412	55	50	⊕	⊕	PROPN
ap-1412	55	51	x	x	PUNCT
ap-1412	56	1	if	if	SCONJ
ap-1412	56	2	one	one	NUM
ap-1412	56	3	side	side	NOUN
ap-1412	56	4	is	be	AUX
ap-1412	56	5	defined	define	VERB
ap-1412	56	6	,	,	PUNCT
ap-1412	56	7	(	(	PUNCT
ap-1412	56	8	ge2	ge2	PROPN
ap-1412	56	9	)	)	PUNCT
ap-1412	56	10	(	(	PUNCT
ap-1412	56	11	x	x	PROPN
ap-1412	56	12	⊕	⊕	PROPN
ap-1412	56	13	y	y	PROPN
ap-1412	56	14	)	)	PUNCT
ap-1412	56	15	⊕	⊕	PROPN
ap-1412	56	16	z	z	PUNCT
ap-1412	57	1	=	=	PUNCT
ap-1412	57	2	x	x	SYM
ap-1412	57	3	⊕	⊕	PROPN
ap-1412	57	4	(	(	PUNCT
ap-1412	57	5	y	y	PROPN
ap-1412	57	6	⊕	⊕	PROPN
ap-1412	57	7	z	z	PROPN
ap-1412	57	8	)	)	PUNCT
ap-1412	57	9	if	if	SCONJ
ap-1412	57	10	one	one	NUM
ap-1412	57	11	side	side	NOUN
ap-1412	57	12	is	be	AUX
ap-1412	57	13	defined	define	VERB
ap-1412	57	14	,	,	PUNCT
ap-1412	57	15	(	(	PUNCT
ap-1412	57	16	ge3	ge3	NOUN
ap-1412	57	17	)	)	PUNCT
ap-1412	57	18	if	if	SCONJ
ap-1412	57	19	x	x	PROPN
ap-1412	57	20	⊕	⊕	NOUN
ap-1412	57	21	y	y	NOUN
ap-1412	57	22	=	=	SYM
ap-1412	57	23	x	x	SYM
ap-1412	57	24	⊕	⊕	PROPN
ap-1412	57	25	z	z	NOUN
ap-1412	58	1	then	then	ADV
ap-1412	58	2	y	y	PROPN
ap-1412	58	3	=	=	SYM
ap-1412	58	4	z	z	PROPN
ap-1412	58	5	,	,	PUNCT
ap-1412	58	6	(	(	PUNCT
ap-1412	58	7	ge4	ge4	PROPN
ap-1412	58	8	)	)	PUNCT
ap-1412	58	9	if	if	SCONJ
ap-1412	58	10	x	x	PROPN
ap-1412	58	11	⊕	⊕	NOUN
ap-1412	58	12	y	y	NOUN
ap-1412	59	1	=	=	PUNCT
ap-1412	59	2	0	0	PUNCT
ap-1412	59	3	then	then	ADV
ap-1412	59	4	x	x	X
ap-1412	59	5	=	=	SYM
ap-1412	59	6	y	y	PROPN
ap-1412	59	7	=	=	SYM
ap-1412	59	8	0	0	PROPN
ap-1412	59	9	,	,	PUNCT
ap-1412	59	10	(	(	PUNCT
ap-1412	59	11	ge5	ge5	PROPN
ap-1412	59	12	)	)	PUNCT
ap-1412	59	13	x	x	PUNCT
ap-1412	59	14	⊕	⊕	NOUN
ap-1412	59	15	0	0	PUNCT
ap-1412	60	1	=	=	PUNCT
ap-1412	60	2	x	x	PUNCT
ap-1412	60	3	for	for	ADP
ap-1412	60	4	all	all	DET
ap-1412	60	5	x	x	SYM
ap-1412	60	6	∈	∈	PROPN
ap-1412	60	7	e.	e.	PROPN
ap-1412	60	8	(	(	PUNCT
ap-1412	60	9	2	2	NUM
ap-1412	60	10	)	)	PUNCT
ap-1412	60	11	a	a	DET
ap-1412	60	12	binary	binary	ADJ
ap-1412	60	13	relation	relation	NOUN
ap-1412	60	14	≤	≤	NOUN
ap-1412	60	15	(	(	PUNCT
ap-1412	60	16	being	be	AUX
ap-1412	60	17	a	a	DET
ap-1412	60	18	partial	partial	ADJ
ap-1412	60	19	order	order	NOUN
ap-1412	60	20	)	)	PUNCT
ap-1412	60	21	on	on	ADP
ap-1412	60	22	e	e	NOUN
ap-1412	60	23	can	can	AUX
ap-1412	60	24	be	be	AUX
ap-1412	60	25	defined	define	VERB
ap-1412	60	26	by	by	ADP
ap-1412	60	27	x	x	SYM
ap-1412	60	28	≤	≤	PROPN
ap-1412	60	29	y	y	PROPN
ap-1412	60	30	iff	iff	PROPN
ap-1412	60	31	for	for	ADP
ap-1412	60	32	some	some	DET
ap-1412	60	33	z	z	NOUN
ap-1412	60	34	∈	∈	PROPN
ap-1412	60	35	e	e	NOUN
ap-1412	60	36	,	,	PUNCT
ap-1412	60	37	x	x	PROPN
ap-1412	60	38	⊕	⊕	NOUN
ap-1412	60	39	z	z	X
ap-1412	60	40	=	=	SYM
ap-1412	60	41	y	y	PROPN
ap-1412	60	42	.	.	PUNCT
ap-1412	61	1	(	(	PUNCT
ap-1412	61	2	3	3	X
ap-1412	61	3	)	)	PUNCT
ap-1412	61	4	q	q	NOUN
ap-1412	62	1	⊆	⊆	NUM
ap-1412	62	2	e	e	NOUN
ap-1412	62	3	is	be	AUX
ap-1412	62	4	called	call	VERB
ap-1412	62	5	a	a	DET
ap-1412	62	6	sub	sub	ADJ
ap-1412	62	7	-	-	ADJ
ap-1412	62	8	generalized	generalized	ADJ
ap-1412	62	9	effect	effect	NOUN
ap-1412	62	10	algebra	algebra	NOUN
ap-1412	62	11	(	(	PUNCT
ap-1412	62	12	sub	sub	ADJ
ap-1412	62	13	-	-	ADJ
ap-1412	62	14	effect	effect	ADJ
ap-1412	62	15	algebra	algebra	NOUN
ap-1412	62	16	)	)	PUNCT
ap-1412	62	17	of	of	ADP
ap-1412	62	18	the	the	DET
ap-1412	62	19	generalized	generalized	ADJ
ap-1412	62	20	effect	effect	NOUN
ap-1412	62	21	algebra	algebra	NOUN
ap-1412	62	22	e	e	X
ap-1412	62	23	(	(	PUNCT
ap-1412	62	24	effect	effect	NOUN
ap-1412	62	25	algebra	algebra	NOUN
ap-1412	62	26	e	e	NOUN
ap-1412	62	27	)	)	PUNCT
ap-1412	62	28	iff	iff	NOUN
ap-1412	62	29	it	it	PRON
ap-1412	62	30	has	have	VERB
ap-1412	62	31	the	the	DET
ap-1412	62	32	following	follow	VERB
ap-1412	62	33	property	property	NOUN
ap-1412	62	34	.	.	PUNCT
ap-1412	63	1	if	if	SCONJ
ap-1412	63	2	at	at	ADV
ap-1412	63	3	least	least	ADV
ap-1412	63	4	two	two	NUM
ap-1412	63	5	of	of	ADP
ap-1412	63	6	elements	element	NOUN
ap-1412	63	7	x	x	X
ap-1412	63	8	,	,	PUNCT
ap-1412	63	9	y	y	PROPN
ap-1412	63	10	,	,	PUNCT
ap-1412	63	11	z	z	NOUN
ap-1412	63	12	∈	∈	PROPN
ap-1412	63	13	e	e	X
ap-1412	63	14	with	with	ADP
ap-1412	63	15	x⊕	x⊕	PROPN
ap-1412	63	16	y	y	PROPN
ap-1412	63	17	=	=	PUNCT
ap-1412	63	18	z	z	NOUN
ap-1412	63	19	are	be	AUX
ap-1412	63	20	in	in	ADP
ap-1412	63	21	q	q	PROPN
ap-1412	63	22	then	then	ADV
ap-1412	63	23	all	all	DET
ap-1412	63	24	x	x	NOUN
ap-1412	63	25	,	,	PUNCT
ap-1412	63	26	y	y	PROPN
ap-1412	63	27	,	,	PUNCT
ap-1412	63	28	z	z	PROPN
ap-1412	63	29	are	be	AUX
ap-1412	63	30	in	in	ADP
ap-1412	63	31	q.	q.	PROPN
ap-1412	63	32	note	note	PROPN
ap-1412	63	33	that	that	SCONJ
ap-1412	63	34	a	a	DET
ap-1412	63	35	sub	sub	ADJ
ap-1412	63	36	-	-	ADJ
ap-1412	63	37	generalized	generalized	ADJ
ap-1412	63	38	effect	effect	NOUN
ap-1412	63	39	algebra	algebra	NOUN
ap-1412	63	40	(	(	PUNCT
ap-1412	63	41	sub	sub	ADJ
ap-1412	63	42	-	-	ADJ
ap-1412	63	43	effect	effect	ADJ
ap-1412	63	44	algebra	algebra	NOUN
ap-1412	63	45	)	)	PUNCT
ap-1412	63	46	q	q	PUNCT
ap-1412	64	1	⊂	⊂	PUNCT
ap-1412	64	2	e	e	X
ap-1412	64	3	is	be	AUX
ap-1412	64	4	a	a	DET
ap-1412	64	5	(	(	PUNCT
ap-1412	64	6	generalized	generalized	ADJ
ap-1412	64	7	)	)	PUNCT
ap-1412	64	8	effect	effect	NOUN
ap-1412	64	9	algebra	algebra	NOUN
ap-1412	64	10	in	in	ADP
ap-1412	64	11	its	its	PRON
ap-1412	64	12	own	own	ADJ
ap-1412	64	13	right	right	NOUN
ap-1412	64	14	.	.	PUNCT
ap-1412	65	1	3	3	NUM
ap-1412	65	2	generalized	generalized	ADJ
ap-1412	65	3	effect	effect	NOUN
ap-1412	65	4	algebras	algebra	NOUN
ap-1412	65	5	of	of	ADP
ap-1412	65	6	positive	positive	ADJ
ap-1412	65	7	operators	operator	NOUN
ap-1412	65	8	on	on	ADP
ap-1412	65	9	a	a	DET
ap-1412	65	10	hilbert	hilbert	NOUN
ap-1412	65	11	space	space	NOUN
ap-1412	65	12	and	and	CCONJ
ap-1412	65	13	their	their	PRON
ap-1412	65	14	sub	sub	ADJ
ap-1412	65	15	-	-	ADJ
ap-1412	65	16	generalized	generalized	ADJ
ap-1412	65	17	effect	effect	NOUN
ap-1412	65	18	algebras	algebra	VERB
ap-1412	65	19	in	in	ADP
ap-1412	65	20	[	[	PUNCT
ap-1412	65	21	9	9	NUM
ap-1412	65	22	]	]	X
ap-1412	65	23	the	the	DET
ap-1412	65	24	following	follow	VERB
ap-1412	65	25	theorem	theorem	NOUN
ap-1412	65	26	on	on	ADP
ap-1412	65	27	positive	positive	ADJ
ap-1412	65	28	linear	linear	PROPN
ap-1412	65	29	operators	operator	NOUN
ap-1412	65	30	with	with	ADP
ap-1412	65	31	common	common	ADJ
ap-1412	65	32	domain	domain	NOUN
ap-1412	65	33	was	be	AUX
ap-1412	65	34	proved	prove	VERB
ap-1412	65	35	:	:	PUNCT
ap-1412	65	36	theorem	theorem	VERB
ap-1412	65	37	3.1	3.1	NUM
ap-1412	66	1	[	[	SYM
ap-1412	66	2	9	9	NUM
ap-1412	66	3	,	,	PUNCT
ap-1412	66	4	theorem	theorem	VERB
ap-1412	66	5	3.1	3.1	NUM
ap-1412	66	6	]	]	PUNCT
ap-1412	66	7	let	let	VERB
ap-1412	66	8	h	h	PRON
ap-1412	66	9	be	be	AUX
ap-1412	66	10	a	a	DET
ap-1412	66	11	complex	complex	ADJ
ap-1412	66	12	hilbert	hilbert	NOUN
ap-1412	66	13	space	space	NOUN
ap-1412	66	14	and	and	CCONJ
ap-1412	66	15	let	let	VERB
ap-1412	66	16	d	d	PRON
ap-1412	66	17	⊆	⊆	NUM
ap-1412	66	18	h	h	NOUN
ap-1412	66	19	be	be	AUX
ap-1412	66	20	a	a	DET
ap-1412	66	21	linear	linear	ADJ
ap-1412	66	22	subspace	subspace	NOUN
ap-1412	66	23	dense	dense	ADJ
ap-1412	66	24	in	in	ADP
ap-1412	66	25	h	h	PROPN
ap-1412	66	26	(	(	PUNCT
ap-1412	66	27	i.e.	i.e.	X
ap-1412	66	28	,	,	PUNCT
ap-1412	66	29	d̄	d̄	NOUN
ap-1412	66	30	=	=	SYM
ap-1412	66	31	h	h	NOUN
ap-1412	66	32	)	)	PUNCT
ap-1412	66	33	.	.	PUNCT
ap-1412	67	1	let	let	VERB
ap-1412	67	2	gd(h	gd(h	NOUN
ap-1412	67	3	)	)	PUNCT
ap-1412	68	1	=	=	PRON
ap-1412	68	2	{	{	PUNCT
ap-1412	68	3	a	a	DET
ap-1412	68	4	:	:	PUNCT
ap-1412	68	5	d	d	X
ap-1412	68	6	→	→	SYM
ap-1412	68	7	h	h	NOUN
ap-1412	68	8	|	|	ADV
ap-1412	68	9	a	a	PRON
ap-1412	68	10	is	be	AUX
ap-1412	68	11	a	a	DET
ap-1412	68	12	positive	positive	ADJ
ap-1412	68	13	linear	linear	NOUN
ap-1412	68	14	operator	operator	NOUN
ap-1412	68	15	defined	define	VERB
ap-1412	68	16	on	on	ADP
ap-1412	68	17	d	d	NOUN
ap-1412	68	18	}	}	PUNCT
ap-1412	68	19	.	.	PUNCT
ap-1412	69	1	then	then	ADV
ap-1412	69	2	(	(	PUNCT
ap-1412	69	3	gd(h);⊕	gd(h);⊕	NOUN
ap-1412	69	4	,	,	PUNCT
ap-1412	69	5	0	0	NUM
ap-1412	69	6	)	)	PUNCT
ap-1412	69	7	is	be	AUX
ap-1412	69	8	a	a	DET
ap-1412	69	9	generalized	generalized	ADJ
ap-1412	69	10	effect	effect	NOUN
ap-1412	69	11	algebra	algebra	NOUN
ap-1412	69	12	where	where	SCONJ
ap-1412	69	13	0	0	NUM
ap-1412	69	14	is	be	AUX
ap-1412	69	15	the	the	DET
ap-1412	69	16	null	null	ADJ
ap-1412	69	17	operator	operator	NOUN
ap-1412	69	18	and	and	CCONJ
ap-1412	69	19	⊕	⊕	PROPN
ap-1412	69	20	is	be	AUX
ap-1412	69	21	the	the	DET
ap-1412	69	22	usual	usual	ADJ
ap-1412	69	23	sum	sum	NOUN
ap-1412	69	24	of	of	ADP
ap-1412	69	25	operators	operator	NOUN
ap-1412	69	26	defined	define	VERB
ap-1412	69	27	on	on	ADP
ap-1412	69	28	d.	d.	PROPN
ap-1412	69	29	in	in	ADP
ap-1412	69	30	this	this	DET
ap-1412	69	31	case	case	NOUN
ap-1412	69	32	⊕	⊕	PROPN
ap-1412	69	33	is	be	AUX
ap-1412	69	34	a	a	DET
ap-1412	69	35	total	total	ADJ
ap-1412	69	36	operation	operation	NOUN
ap-1412	69	37	.	.	PUNCT
ap-1412	70	1	if	if	SCONJ
ap-1412	70	2	d	d	NOUN
ap-1412	70	3	=	=	SYM
ap-1412	70	4	h	h	NOUN
ap-1412	70	5	in	in	ADP
ap-1412	70	6	theorem	theorem	ADJ
ap-1412	70	7	3.1	3.1	NUM
ap-1412	70	8	then	then	ADV
ap-1412	70	9	gd(h	gd(h	PROPN
ap-1412	70	10	)	)	PUNCT
ap-1412	70	11	is	be	AUX
ap-1412	70	12	a	a	DET
ap-1412	70	13	generalized	generalized	ADJ
ap-1412	70	14	effect	effect	NOUN
ap-1412	70	15	algebra	algebra	NOUN
ap-1412	70	16	of	of	ADP
ap-1412	70	17	all	all	DET
ap-1412	70	18	bounded	bound	VERB
ap-1412	70	19	positive	positive	ADJ
ap-1412	70	20	linear	linear	PROPN
ap-1412	70	21	operators	operator	NOUN
ap-1412	70	22	acting	act	VERB
ap-1412	70	23	in	in	ADP
ap-1412	70	24	h	h	NOUN
ap-1412	70	25	with	with	ADP
ap-1412	70	26	usual	usual	ADJ
ap-1412	70	27	addition	addition	NOUN
ap-1412	70	28	as	as	ADP
ap-1412	70	29	effect	effect	NOUN
ap-1412	70	30	algebraic	algebraic	ADJ
ap-1412	70	31	operation	operation	NOUN
ap-1412	70	32	⊕.	⊕.	NOUN
ap-1412	70	33	hence	hence	ADV
ap-1412	70	34	in	in	ADP
ap-1412	70	35	the	the	DET
ap-1412	70	36	case	case	NOUN
ap-1412	70	37	d	d	NOUN
ap-1412	70	38	=	=	SYM
ap-1412	70	39	h	h	NOUN
ap-1412	70	40	all	all	DET
ap-1412	70	41	operators	operator	NOUN
ap-1412	70	42	in	in	ADP
ap-1412	70	43	gd(h	gd(h	PROPN
ap-1412	70	44	)	)	PUNCT
ap-1412	70	45	are	be	AUX
ap-1412	70	46	self	self	NOUN
ap-1412	70	47	-	-	PUNCT
ap-1412	70	48	adjoint	adjoint	NOUN
ap-1412	70	49	.	.	PUNCT
ap-1412	71	1	on	on	ADP
ap-1412	71	2	the	the	DET
ap-1412	71	3	other	other	ADJ
ap-1412	71	4	hand	hand	NOUN
ap-1412	71	5	,	,	PUNCT
ap-1412	71	6	if	if	SCONJ
ap-1412	71	7	d	d	PROPN
ap-1412	71	8	�	�	PROPN
ap-1412	71	9	=	=	SYM
ap-1412	71	10	h	h	NOUN
ap-1412	71	11	then	then	ADV
ap-1412	71	12	every	every	DET
ap-1412	71	13	bounded	bounded	ADJ
ap-1412	71	14	operator	operator	NOUN
ap-1412	71	15	in	in	ADP
ap-1412	71	16	gd(h	gd(h	PROPN
ap-1412	71	17	)	)	PUNCT
ap-1412	71	18	is	be	AUX
ap-1412	71	19	a	a	DET
ap-1412	71	20	restriction	restriction	NOUN
ap-1412	71	21	a|d	a|d	PROPN
ap-1412	71	22	of	of	ADP
ap-1412	71	23	a	a	DET
ap-1412	71	24	bounded	bound	VERB
ap-1412	71	25	operator	operator	NOUN
ap-1412	71	26	a	a	PRON
ap-1412	71	27	with	with	ADP
ap-1412	71	28	d(a	d(a	PROPN
ap-1412	71	29	)	)	PUNCT
ap-1412	71	30	=	=	SYM
ap-1412	72	1	h.	h.	PROPN
ap-1412	72	2	thus	thus	ADV
ap-1412	72	3	,	,	PUNCT
ap-1412	72	4	in	in	ADP
ap-1412	72	5	this	this	DET
ap-1412	72	6	case	case	NOUN
ap-1412	72	7	,	,	PUNCT
ap-1412	72	8	a	a	DET
ap-1412	72	9	=	=	NOUN
ap-1412	72	10	a∗	a∗	NOUN
ap-1412	72	11	=	=	SYM
ap-1412	72	12	(	(	PUNCT
ap-1412	72	13	a|d)∗	a|d)∗	PROPN
ap-1412	72	14	�	�	PROPN
ap-1412	72	15	=	=	PUNCT
ap-1412	72	16	a|d	a|d	PROPN
ap-1412	72	17	.	.	PUNCT
ap-1412	73	1	it	it	PRON
ap-1412	73	2	follows	follow	VERB
ap-1412	73	3	that	that	SCONJ
ap-1412	73	4	every	every	DET
ap-1412	73	5	79	79	NUM
ap-1412	73	6	acta	acta	PROPN
ap-1412	73	7	polytechnica	polytechnica	PROPN
ap-1412	73	8	vol	vol	NOUN
ap-1412	73	9	.	.	PUNCT
ap-1412	74	1	51	51	NUM
ap-1412	74	2	no	no	NOUN
ap-1412	74	3	.	.	PUNCT
ap-1412	75	1	4/2011	4/2011	NUM
ap-1412	75	2	self	self	NOUN
ap-1412	75	3	-	-	PUNCT
ap-1412	75	4	adjoint	adjoint	NOUN
ap-1412	75	5	operator	operator	NOUN
ap-1412	75	6	in	in	ADP
ap-1412	75	7	gd(h	gd(h	PROPN
ap-1412	75	8	)	)	PUNCT
ap-1412	75	9	for	for	ADP
ap-1412	75	10	d	d	PROPN
ap-1412	75	11	�	�	PROPN
ap-1412	75	12	=	=	NOUN
ap-1412	75	13	h	h	NOUN
ap-1412	75	14	is	be	AUX
ap-1412	75	15	necessarily	necessarily	ADV
ap-1412	75	16	unbounded	unbounded	ADJ
ap-1412	75	17	.	.	PUNCT
ap-1412	76	1	nevertheless	nevertheless	ADV
ap-1412	76	2	,	,	PUNCT
ap-1412	76	3	it	it	PRON
ap-1412	76	4	is	be	AUX
ap-1412	76	5	well	well	ADV
ap-1412	76	6	known	know	VERB
ap-1412	76	7	(	(	PUNCT
ap-1412	76	8	see	see	VERB
ap-1412	76	9	,	,	PUNCT
ap-1412	76	10	e.g.	e.g.	ADV
ap-1412	76	11	[	[	X
ap-1412	76	12	10	10	NUM
ap-1412	76	13	]	]	PUNCT
ap-1412	76	14	)	)	PUNCT
ap-1412	76	15	that	that	SCONJ
ap-1412	76	16	every	every	DET
ap-1412	76	17	densely	densely	ADV
ap-1412	76	18	defined	define	VERB
ap-1412	76	19	positive	positive	ADJ
ap-1412	76	20	operator	operator	NOUN
ap-1412	76	21	a	a	PRON
ap-1412	76	22	has	have	AUX
ap-1412	76	23	a	a	DET
ap-1412	76	24	positive	positive	ADJ
ap-1412	76	25	self	self	NOUN
ap-1412	76	26	-	-	PUNCT
ap-1412	76	27	adjoint	adjoint	NOUN
ap-1412	76	28	extension	extension	NOUN
ap-1412	76	29	â	â	ADP
ap-1412	76	30	called	call	VERB
ap-1412	76	31	friedrichs	friedrich	NOUN
ap-1412	76	32	’	'	PUNCT
ap-1412	76	33	extension	extension	NOUN
ap-1412	76	34	.	.	PUNCT
ap-1412	77	1	moreover	moreover	ADV
ap-1412	77	2	,	,	PUNCT
ap-1412	77	3	â	â	X
ap-1412	77	4	extends	extend	VERB
ap-1412	77	5	all	all	DET
ap-1412	77	6	symmetric	symmetric	ADJ
ap-1412	77	7	extension	extension	NOUN
ap-1412	77	8	a′	a′	PROPN
ap-1412	77	9	of	of	ADP
ap-1412	77	10	a.	a.	NOUN
ap-1412	78	1	thus	thus	ADV
ap-1412	78	2	if	if	SCONJ
ap-1412	78	3	a′	a′	PROPN
ap-1412	78	4	is	be	AUX
ap-1412	78	5	selfadjoint	selfadjoint	NOUN
ap-1412	78	6	then	then	ADV
ap-1412	78	7	a′	a′	PROPN
ap-1412	79	1	=	=	SYM
ap-1412	79	2	â.	â.	ADJ
ap-1412	79	3	but	but	CCONJ
ap-1412	79	4	in	in	ADP
ap-1412	79	5	general	general	ADJ
ap-1412	79	6	d(a	d(a	PROPN
ap-1412	79	7	)	)	PUNCT
ap-1412	79	8	�	�	PROPN
ap-1412	79	9	=	=	SYM
ap-1412	79	10	d(â	d(â	NOUN
ap-1412	79	11	)	)	PUNCT
ap-1412	79	12	,	,	PUNCT
ap-1412	79	13	hence	hence	ADV
ap-1412	79	14	â	â	PROPN
ap-1412	79	15	/∈	/∈	PUNCT
ap-1412	79	16	gd(h	gd(h	NOUN
ap-1412	79	17	)	)	PUNCT
ap-1412	79	18	.	.	PUNCT
ap-1412	80	1	clearly	clearly	ADV
ap-1412	80	2	,	,	PUNCT
ap-1412	80	3	for	for	ADP
ap-1412	80	4	domains	domain	NOUN
ap-1412	80	5	d1	d1	PROPN
ap-1412	80	6	�	�	PROPN
ap-1412	80	7	=	=	SYM
ap-1412	80	8	d2	d2	PROPN
ap-1412	80	9	,	,	PUNCT
ap-1412	80	10	gd1	gd1	PROPN
ap-1412	80	11	(	(	PUNCT
ap-1412	80	12	h)∩gd2	h)∩gd2	PROPN
ap-1412	80	13	(	(	PUNCT
ap-1412	80	14	h	h	NOUN
ap-1412	80	15	)	)	PUNCT
ap-1412	80	16	=	=	PUNCT
ap-1412	80	17	∅.	∅.	NOUN
ap-1412	80	18	however	however	ADV
ap-1412	80	19	it	it	PRON
ap-1412	80	20	is	be	AUX
ap-1412	80	21	well	well	ADV
ap-1412	80	22	-	-	PUNCT
ap-1412	80	23	known	know	VERB
ap-1412	80	24	that	that	PRON
ap-1412	80	25	bounded	bound	VERB
ap-1412	80	26	linear	linear	PROPN
ap-1412	80	27	operators	operator	NOUN
ap-1412	80	28	have	have	VERB
ap-1412	80	29	unique	unique	ADJ
ap-1412	80	30	extensions	extension	NOUN
ap-1412	80	31	to	to	ADP
ap-1412	80	32	the	the	DET
ap-1412	80	33	whole	whole	ADJ
ap-1412	80	34	space	space	NOUN
ap-1412	80	35	h.	h.	PROPN
ap-1412	80	36	theorem	theorem	VERB
ap-1412	80	37	3.1	3.1	NUM
ap-1412	80	38	remains	remain	VERB
ap-1412	80	39	true	true	ADJ
ap-1412	80	40	if	if	SCONJ
ap-1412	80	41	we	we	PRON
ap-1412	80	42	substitute	substitute	VERB
ap-1412	80	43	gd(h	gd(h	NOUN
ap-1412	80	44	)	)	PUNCT
ap-1412	80	45	by	by	ADP
ap-1412	80	46	g̃d(h	g̃d(h	PROPN
ap-1412	80	47	)	)	PUNCT
ap-1412	81	1	=	=	PRON
ap-1412	81	2	{	{	PUNCT
ap-1412	81	3	a	a	X
ap-1412	81	4	:	:	PUNCT
ap-1412	81	5	d(a	d(a	PROPN
ap-1412	81	6	)	)	PUNCT
ap-1412	81	7	→	→	SYM
ap-1412	81	8	h	h	NOUN
ap-1412	82	1	|	|	ADV
ap-1412	82	2	a	a	PRON
ap-1412	82	3	is	be	AUX
ap-1412	82	4	positive	positive	ADJ
ap-1412	82	5	linear	linear	ADJ
ap-1412	82	6	operator	operator	NOUN
ap-1412	82	7	with	with	ADP
ap-1412	82	8	d(a	d(a	PROPN
ap-1412	82	9	)	)	PUNCT
ap-1412	83	1	=	=	PUNCT
ap-1412	84	1	d	d	NOUN
ap-1412	84	2	if	if	SCONJ
ap-1412	84	3	a	a	PRON
ap-1412	84	4	is	be	AUX
ap-1412	84	5	unbounded	unbounded	ADJ
ap-1412	84	6	,	,	PUNCT
ap-1412	84	7	d(a	d(a	PROPN
ap-1412	84	8	)	)	PUNCT
ap-1412	85	1	=	=	SYM
ap-1412	85	2	h	h	NOUN
ap-1412	85	3	if	if	SCONJ
ap-1412	85	4	it	it	PRON
ap-1412	85	5	is	be	AUX
ap-1412	85	6	bounded	bound	VERB
ap-1412	85	7	}	}	PUNCT
ap-1412	85	8	.	.	PUNCT
ap-1412	86	1	then	then	ADV
ap-1412	86	2	for	for	ADP
ap-1412	86	3	d1	d1	PROPN
ap-1412	86	4	�	�	PROPN
ap-1412	86	5	=	=	SYM
ap-1412	86	6	d2	d2	PROPN
ap-1412	86	7	we	we	PRON
ap-1412	86	8	obtain	obtain	VERB
ap-1412	86	9	g̃d1	g̃d1	NOUN
ap-1412	86	10	(	(	PUNCT
ap-1412	86	11	h	h	NOUN
ap-1412	86	12	)	)	PUNCT
ap-1412	86	13	∩	∩	ADJ
ap-1412	86	14	g̃d2(h	g̃d2(h	PROPN
ap-1412	86	15	)	)	PUNCT
ap-1412	86	16	=	=	SYM
ap-1412	86	17	b+(h	b+(h	PROPN
ap-1412	86	18	)	)	PUNCT
ap-1412	86	19	where	where	SCONJ
ap-1412	86	20	b+(h	b+(h	NUM
ap-1412	86	21	)	)	PUNCT
ap-1412	86	22	is	be	AUX
ap-1412	86	23	the	the	DET
ap-1412	86	24	set	set	NOUN
ap-1412	86	25	of	of	ADP
ap-1412	86	26	all	all	DET
ap-1412	86	27	bounded	bound	VERB
ap-1412	86	28	positive	positive	ADJ
ap-1412	86	29	linear	linear	PROPN
ap-1412	86	30	operators	operator	NOUN
ap-1412	86	31	a	a	PRON
ap-1412	86	32	with	with	ADP
ap-1412	86	33	d(a	d(a	PROPN
ap-1412	86	34	)	)	PUNCT
ap-1412	86	35	=	=	SYM
ap-1412	86	36	h.	h.	PROPN
ap-1412	86	37	theorem	theorem	VERB
ap-1412	86	38	3.2	3.2	NUM
ap-1412	87	1	[	[	SYM
ap-1412	87	2	9	9	NUM
ap-1412	87	3	,	,	PUNCT
ap-1412	87	4	theorem	theorem	VERB
ap-1412	87	5	3.5	3.5	NUM
ap-1412	87	6	]	]	PUNCT
ap-1412	87	7	let	let	VERB
ap-1412	87	8	h	h	PRON
ap-1412	87	9	be	be	AUX
ap-1412	87	10	an	an	DET
ap-1412	87	11	infinite	infinite	ADJ
ap-1412	87	12	-	-	PUNCT
ap-1412	87	13	dimensional	dimensional	ADJ
ap-1412	87	14	complex	complex	ADJ
ap-1412	87	15	hilbert	hilbert	NOUN
ap-1412	87	16	space	space	NOUN
ap-1412	87	17	.	.	PUNCT
ap-1412	88	1	let	let	VERB
ap-1412	88	2	v(h	v(h	NOUN
ap-1412	88	3	)	)	PUNCT
ap-1412	89	1	=	=	PRON
ap-1412	89	2	{	{	PUNCT
ap-1412	89	3	a	a	X
ap-1412	89	4	:	:	PUNCT
ap-1412	89	5	d(a	d(a	PROPN
ap-1412	89	6	)	)	PUNCT
ap-1412	89	7	→	→	SYM
ap-1412	89	8	h	h	NOUN
ap-1412	89	9	|	|	ADV
ap-1412	89	10	a	a	DET
ap-1412	89	11	≥	≥	NOUN
ap-1412	89	12	0	0	NUM
ap-1412	89	13	with	with	ADP
ap-1412	89	14	d(a	d(a	PROPN
ap-1412	89	15	)	)	PUNCT
ap-1412	89	16	=	=	SYM
ap-1412	89	17	h	h	NOUN
ap-1412	89	18	and	and	CCONJ
ap-1412	89	19	d(a	d(a	PROPN
ap-1412	89	20	)	)	PUNCT
ap-1412	90	1	=	=	SYM
ap-1412	90	2	h	h	NOUN
ap-1412	90	3	if	if	SCONJ
ap-1412	90	4	a	a	PRON
ap-1412	90	5	is	be	AUX
ap-1412	90	6	bounded	bound	VERB
ap-1412	90	7	}	}	PUNCT
ap-1412	90	8	.	.	PUNCT
ap-1412	91	1	let	let	VERB
ap-1412	91	2	⊕	⊕	PROPN
ap-1412	91	3	be	be	AUX
ap-1412	91	4	a	a	DET
ap-1412	91	5	partial	partial	ADJ
ap-1412	91	6	binary	binary	ADJ
ap-1412	91	7	operation	operation	NOUN
ap-1412	91	8	on	on	ADP
ap-1412	91	9	v(h	v(h	NOUN
ap-1412	91	10	)	)	PUNCT
ap-1412	91	11	defined	define	VERB
ap-1412	91	12	by	by	ADP
ap-1412	91	13	a⊕b	a⊕b	NOUN
ap-1412	91	14	=	=	SYM
ap-1412	91	15	a+b	a+b	NUM
ap-1412	91	16	with	with	ADP
ap-1412	91	17	d(a⊕b	d(a⊕b	PROPN
ap-1412	91	18	)	)	PUNCT
ap-1412	92	1	=	=	SYM
ap-1412	92	2	h	h	NOUN
ap-1412	92	3	for	for	ADP
ap-1412	92	4	any	any	DET
ap-1412	92	5	bounded	bounded	ADJ
ap-1412	92	6	a	a	PRON
ap-1412	92	7	,	,	PUNCT
ap-1412	92	8	b	b	PROPN
ap-1412	92	9	∈	∈	PROPN
ap-1412	92	10	v(h	v(h	NOUN
ap-1412	92	11	)	)	PUNCT
ap-1412	92	12	and	and	CCONJ
ap-1412	92	13	a	a	DET
ap-1412	92	14	⊕	⊕	PROPN
ap-1412	92	15	b	b	PROPN
ap-1412	92	16	=	=	SYM
ap-1412	92	17	b	b	PROPN
ap-1412	92	18	⊕	⊕	PROPN
ap-1412	92	19	a	a	X
ap-1412	92	20	=	=	X
ap-1412	92	21	a	a	DET
ap-1412	92	22	+	+	ADV
ap-1412	92	23	b|d(a	b|d(a	NOUN
ap-1412	92	24	)	)	PUNCT
ap-1412	92	25	with	with	ADP
ap-1412	92	26	d(a	d(a	PROPN
ap-1412	92	27	⊕	⊕	PROPN
ap-1412	92	28	b	b	PROPN
ap-1412	92	29	)	)	PUNCT
ap-1412	92	30	=	=	SYM
ap-1412	93	1	d(a	d(a	PROPN
ap-1412	93	2	)	)	PUNCT
ap-1412	93	3	if	if	SCONJ
ap-1412	93	4	a	a	PRON
ap-1412	93	5	is	be	AUX
ap-1412	93	6	unbounded	unbounded	ADJ
ap-1412	93	7	and	and	CCONJ
ap-1412	93	8	b	b	NOUN
ap-1412	93	9	is	be	AUX
ap-1412	93	10	bounded	bound	VERB
ap-1412	93	11	.	.	PUNCT
ap-1412	94	1	then	then	ADV
ap-1412	94	2	(	(	PUNCT
ap-1412	94	3	v(h);⊕	v(h);⊕	VERB
ap-1412	94	4	,	,	PUNCT
ap-1412	94	5	0	0	NUM
ap-1412	94	6	)	)	PUNCT
ap-1412	94	7	is	be	AUX
ap-1412	94	8	a	a	DET
ap-1412	94	9	generalized	generalized	ADJ
ap-1412	94	10	effect	effect	NOUN
ap-1412	94	11	algebra	algebra	NOUN
ap-1412	94	12	.	.	PUNCT
ap-1412	95	1	moreover	moreover	ADV
ap-1412	95	2	,	,	PUNCT
ap-1412	95	3	b+(h	b+(h	PROPN
ap-1412	95	4	)	)	PUNCT
ap-1412	95	5	is	be	AUX
ap-1412	95	6	a	a	DET
ap-1412	95	7	sub	sub	ADJ
ap-1412	95	8	-	-	ADJ
ap-1412	95	9	generalized	generalized	ADJ
ap-1412	95	10	effect	effect	NOUN
ap-1412	95	11	algebra	algebra	NOUN
ap-1412	95	12	of	of	ADP
ap-1412	95	13	v(h	v(h	NOUN
ap-1412	95	14	)	)	PUNCT
ap-1412	95	15	with	with	ADP
ap-1412	95	16	respect	respect	NOUN
ap-1412	95	17	to	to	ADP
ap-1412	95	18	inherited	inherit	VERB
ap-1412	95	19	⊕-operation	⊕-operation	NOUN
ap-1412	95	20	,	,	PUNCT
ap-1412	95	21	which	which	PRON
ap-1412	95	22	is	be	AUX
ap-1412	95	23	defined	define	VERB
ap-1412	95	24	for	for	ADP
ap-1412	95	25	every	every	DET
ap-1412	95	26	pair	pair	NOUN
ap-1412	95	27	a	a	ADP
ap-1412	95	28	,	,	PUNCT
ap-1412	95	29	b	b	PROPN
ap-1412	95	30	∈	∈	PROPN
ap-1412	95	31	b+(h	b+(h	PROPN
ap-1412	95	32	)	)	PUNCT
ap-1412	95	33	.	.	PUNCT
ap-1412	96	1	now	now	ADV
ap-1412	96	2	we	we	PRON
ap-1412	96	3	are	be	AUX
ap-1412	96	4	going	go	VERB
ap-1412	96	5	to	to	PART
ap-1412	96	6	show	show	VERB
ap-1412	96	7	that	that	SCONJ
ap-1412	96	8	sp(h	sp(h	NOUN
ap-1412	96	9	)	)	PUNCT
ap-1412	96	10	=	=	SYM
ap-1412	96	11	{	{	PUNCT
ap-1412	96	12	a	a	DET
ap-1412	96	13	∈	∈	PROPN
ap-1412	96	14	v(h	v(h	NOUN
ap-1412	96	15	)	)	PUNCT
ap-1412	96	16	|	|	ADV
ap-1412	96	17	a	a	DET
ap-1412	96	18	=	=	NOUN
ap-1412	96	19	a∗	a∗	NOUN
ap-1412	96	20	}	}	PUNCT
ap-1412	96	21	is	be	AUX
ap-1412	96	22	a	a	DET
ap-1412	96	23	sub	sub	ADJ
ap-1412	96	24	-	-	ADJ
ap-1412	96	25	generalized	generalized	ADJ
ap-1412	96	26	effect	effect	NOUN
ap-1412	96	27	algebra	algebra	NOUN
ap-1412	96	28	of	of	ADP
ap-1412	96	29	v(h	v(h	NOUN
ap-1412	96	30	)	)	PUNCT
ap-1412	96	31	,	,	PUNCT
ap-1412	96	32	hence	hence	ADV
ap-1412	96	33	it	it	PRON
ap-1412	96	34	is	be	AUX
ap-1412	96	35	a	a	DET
ap-1412	96	36	generalized	generalized	ADJ
ap-1412	96	37	effect	effect	NOUN
ap-1412	96	38	algebra	algebra	NOUN
ap-1412	96	39	.	.	PUNCT
ap-1412	97	1	moreover	moreover	ADV
ap-1412	97	2	,	,	PUNCT
ap-1412	97	3	we	we	PRON
ap-1412	97	4	show	show	VERB
ap-1412	97	5	that	that	SCONJ
ap-1412	97	6	sp(h	sp(h	NOUN
ap-1412	97	7	)	)	PUNCT
ap-1412	97	8	=	=	SYM
ap-1412	97	9	f(h	f(h	NOUN
ap-1412	97	10	)	)	PUNCT
ap-1412	97	11	=	=	PRON
ap-1412	97	12	{	{	PUNCT
ap-1412	97	13	â	â	X
ap-1412	97	14	|	|	ADV
ap-1412	97	15	a	a	DET
ap-1412	97	16	∈	∈	PROPN
ap-1412	97	17	v(h	v(h	NOUN
ap-1412	97	18	)	)	PUNCT
ap-1412	97	19	,	,	PUNCT
ap-1412	97	20	â	â	X
ap-1412	97	21	is	be	AUX
ap-1412	97	22	a	a	DET
ap-1412	97	23	friedrichs	friedrich	NOUN
ap-1412	97	24	positive	positive	ADJ
ap-1412	97	25	self	self	NOUN
ap-1412	97	26	-	-	PUNCT
ap-1412	97	27	adjoint	adjoint	NOUN
ap-1412	97	28	extension	extension	NOUN
ap-1412	97	29	of	of	ADP
ap-1412	97	30	a	a	PRON
ap-1412	97	31	}	}	PUNCT
ap-1412	97	32	.	.	PUNCT
ap-1412	98	1	lemma	lemma	PROPN
ap-1412	98	2	3.3	3.3	NUM
ap-1412	98	3	under	under	ADP
ap-1412	98	4	the	the	DET
ap-1412	98	5	assumptions	assumption	NOUN
ap-1412	98	6	of	of	ADP
ap-1412	98	7	theorem	theorem	NOUN
ap-1412	98	8	3.2	3.2	NUM
ap-1412	98	9	,	,	PUNCT
ap-1412	98	10	for	for	ADP
ap-1412	98	11	every	every	DET
ap-1412	98	12	a	a	DET
ap-1412	98	13	∈	∈	PROPN
ap-1412	98	14	v(h	v(h	NOUN
ap-1412	98	15	):	):	PUNCT
ap-1412	98	16	(	(	PUNCT
ap-1412	98	17	i	i	NOUN
ap-1412	98	18	)	)	PUNCT
ap-1412	98	19	a	a	PRON
ap-1412	98	20	and	and	CCONJ
ap-1412	98	21	a∗	a∗	PROPN
ap-1412	98	22	exist	exist	NOUN
ap-1412	98	23	,	,	PUNCT
ap-1412	98	24	â	â	X
ap-1412	98	25	is	be	AUX
ap-1412	98	26	closed	close	VERB
ap-1412	98	27	and	and	CCONJ
ap-1412	98	28	a	a	DET
ap-1412	98	29	⊂	⊂	PROPN
ap-1412	98	30	a	a	X
ap-1412	98	31	⊂	⊂	X
ap-1412	98	32	â	â	X
ap-1412	99	1	=	=	SYM
ap-1412	99	2	(	(	PUNCT
ap-1412	99	3	â)∗	â)∗	PROPN
ap-1412	99	4	⊂	⊂	X
ap-1412	99	5	(	(	PUNCT
ap-1412	99	6	a)∗	a)∗	PROPN
ap-1412	99	7	=	=	SYM
ap-1412	99	8	a∗	a∗	PROPN
ap-1412	99	9	,	,	PUNCT
ap-1412	99	10	(	(	PUNCT
ap-1412	99	11	ii	ii	NOUN
ap-1412	99	12	)	)	PUNCT
ap-1412	99	13	â	â	PUNCT
ap-1412	100	1	=	=	PUNCT
ap-1412	100	2	a	a	DET
ap-1412	100	3	iff	iff	PROPN
ap-1412	100	4	a	a	PRON
ap-1412	100	5	is	be	AUX
ap-1412	100	6	essentially	essentially	ADV
ap-1412	100	7	self	self	NOUN
ap-1412	100	8	-	-	PUNCT
ap-1412	100	9	adjoint	adjoint	NOUN
ap-1412	100	10	,	,	PUNCT
ap-1412	100	11	(	(	PUNCT
ap-1412	100	12	iii	iii	NOUN
ap-1412	100	13	)	)	PUNCT
ap-1412	100	14	sp(h	sp(h	NOUN
ap-1412	100	15	)	)	PUNCT
ap-1412	100	16	=	=	SYM
ap-1412	100	17	f(h	f(h	PROPN
ap-1412	100	18	)	)	PUNCT
ap-1412	100	19	.	.	PUNCT
ap-1412	101	1	proof	proof	NOUN
ap-1412	101	2	.	.	PUNCT
ap-1412	102	1	(	(	PUNCT
ap-1412	102	2	i	i	NOUN
ap-1412	102	3	)	)	PUNCT
ap-1412	102	4	let	let	VERB
ap-1412	102	5	a	a	DET
ap-1412	102	6	∈	∈	PROPN
ap-1412	102	7	v(h	v(h	NOUN
ap-1412	102	8	)	)	PUNCT
ap-1412	102	9	.	.	PUNCT
ap-1412	103	1	since	since	SCONJ
ap-1412	103	2	d(a	d(a	PROPN
ap-1412	103	3	)	)	PUNCT
ap-1412	103	4	=	=	SYM
ap-1412	103	5	h	h	NOUN
ap-1412	103	6	,	,	PUNCT
ap-1412	103	7	the	the	DET
ap-1412	103	8	adjoint	adjoint	NOUN
ap-1412	103	9	a∗	a∗	NOUN
ap-1412	103	10	of	of	ADP
ap-1412	103	11	a	a	DET
ap-1412	103	12	exists	exist	NOUN
ap-1412	103	13	[	[	X
ap-1412	103	14	3	3	NUM
ap-1412	103	15	,	,	PUNCT
ap-1412	103	16	p.	p.	NOUN
ap-1412	103	17	93	93	NUM
ap-1412	103	18	]	]	PUNCT
ap-1412	103	19	.	.	PUNCT
ap-1412	104	1	further	far	ADV
ap-1412	104	2	the	the	DET
ap-1412	104	3	assumption	assumption	NOUN
ap-1412	104	4	that	that	SCONJ
ap-1412	104	5	a	a	DET
ap-1412	104	6	≥	≥	NOUN
ap-1412	104	7	0	0	NUM
ap-1412	104	8	implies	imply	VERB
ap-1412	104	9	that	that	SCONJ
ap-1412	104	10	a	a	PRON
ap-1412	104	11	is	be	AUX
ap-1412	104	12	symmetric	symmetric	ADJ
ap-1412	104	13	(	(	PUNCT
ap-1412	104	14	see	see	VERB
ap-1412	104	15	[	[	X
ap-1412	104	16	3	3	NUM
ap-1412	104	17	,	,	PUNCT
ap-1412	104	18	p.	p.	NOUN
ap-1412	104	19	142	142	NUM
ap-1412	104	20	]	]	PUNCT
ap-1412	104	21	)	)	PUNCT
ap-1412	104	22	and	and	CCONJ
ap-1412	104	23	there	there	PRON
ap-1412	104	24	exists	exist	VERB
ap-1412	104	25	the	the	DET
ap-1412	104	26	so	so	ADV
ap-1412	104	27	-	-	PUNCT
ap-1412	104	28	called	call	VERB
ap-1412	104	29	friedrichs	friedrich	NOUN
ap-1412	104	30	positive	positive	ADJ
ap-1412	104	31	self	self	NOUN
ap-1412	104	32	-	-	PUNCT
ap-1412	104	33	adjoint	adjoint	NOUN
ap-1412	104	34	extension	extension	NOUN
ap-1412	104	35	â	â	X
ap-1412	104	36	of	of	ADP
ap-1412	104	37	a	a	DET
ap-1412	104	38	(	(	PUNCT
ap-1412	104	39	see	see	NOUN
ap-1412	104	40	,	,	PUNCT
ap-1412	105	1	e.g.	e.g.	ADV
ap-1412	105	2	[	[	X
ap-1412	105	3	3	3	NUM
ap-1412	105	4	]	]	PUNCT
ap-1412	105	5	or	or	CCONJ
ap-1412	105	6	[	[	X
ap-1412	105	7	10	10	NUM
ap-1412	105	8	]	]	NUM
ap-1412	105	9	)	)	PUNCT
ap-1412	105	10	,	,	PUNCT
ap-1412	105	11	hence	hence	ADV
ap-1412	105	12	a	a	DET
ap-1412	105	13	⊂	⊂	X
ap-1412	105	14	â	â	X
ap-1412	105	15	=	=	SYM
ap-1412	105	16	(	(	PUNCT
ap-1412	105	17	â)∗	â)∗	PROPN
ap-1412	105	18	⊂	⊂	ADJ
ap-1412	105	19	a∗.	a∗.	VERB
ap-1412	105	20	it	it	PRON
ap-1412	105	21	follows	follow	VERB
ap-1412	105	22	that	that	SCONJ
ap-1412	105	23	h	h	NOUN
ap-1412	105	24	=	=	SYM
ap-1412	105	25	d(a	d(a	PROPN
ap-1412	105	26	)	)	PUNCT
ap-1412	105	27	=	=	SYM
ap-1412	105	28	d(â	d(â	NOUN
ap-1412	105	29	)	)	PUNCT
ap-1412	105	30	=	=	SYM
ap-1412	105	31	d(a∗	d(a∗	NUM
ap-1412	105	32	)	)	PUNCT
ap-1412	105	33	,	,	PUNCT
ap-1412	105	34	which	which	PRON
ap-1412	105	35	gives	give	VERB
ap-1412	105	36	that	that	DET
ap-1412	105	37	a∗	a∗	NOUN
ap-1412	105	38	and	and	CCONJ
ap-1412	105	39	(	(	PUNCT
ap-1412	105	40	â)∗	â)∗	ADV
ap-1412	105	41	are	be	AUX
ap-1412	105	42	closed	close	VERB
ap-1412	105	43	(	(	PUNCT
ap-1412	105	44	see	see	VERB
ap-1412	105	45	[	[	X
ap-1412	105	46	3	3	NUM
ap-1412	105	47	,	,	PUNCT
ap-1412	105	48	p.	p.	NOUN
ap-1412	105	49	95	95	NUM
ap-1412	105	50	]	]	PUNCT
ap-1412	105	51	)	)	PUNCT
ap-1412	105	52	.	.	PUNCT
ap-1412	106	1	as	as	ADP
ap-1412	106	2	â	â	PROPN
ap-1412	106	3	=	=	PUNCT
ap-1412	106	4	(	(	PUNCT
ap-1412	106	5	â)∗	â)∗	ADV
ap-1412	106	6	,	,	PUNCT
ap-1412	106	7	we	we	PRON
ap-1412	106	8	obtain	obtain	VERB
ap-1412	106	9	that	that	PRON
ap-1412	106	10	â	â	PRON
ap-1412	106	11	is	be	AUX
ap-1412	106	12	closed	closed	ADJ
ap-1412	106	13	.	.	PUNCT
ap-1412	107	1	moreover	moreover	ADV
ap-1412	107	2	,	,	PUNCT
ap-1412	107	3	since	since	SCONJ
ap-1412	107	4	a	a	PRON
ap-1412	107	5	is	be	AUX
ap-1412	107	6	symmetric	symmetric	ADJ
ap-1412	107	7	,	,	PUNCT
ap-1412	107	8	its	its	PRON
ap-1412	107	9	closure	closure	NOUN
ap-1412	107	10	a	a	PRON
ap-1412	107	11	is	be	AUX
ap-1412	107	12	also	also	ADV
ap-1412	107	13	symmetric	symmetric	ADJ
ap-1412	107	14	(	(	PUNCT
ap-1412	107	15	see	see	VERB
ap-1412	107	16	[	[	X
ap-1412	107	17	3	3	NUM
ap-1412	107	18	,	,	PUNCT
ap-1412	107	19	p.	p.	NOUN
ap-1412	107	20	96	96	NUM
ap-1412	107	21	]	]	PUNCT
ap-1412	107	22	)	)	PUNCT
ap-1412	107	23	.	.	PUNCT
ap-1412	108	1	thus	thus	ADV
ap-1412	108	2	we	we	PRON
ap-1412	108	3	obtain	obtain	VERB
ap-1412	108	4	a	a	DET
ap-1412	108	5	⊂	⊂	PROPN
ap-1412	108	6	a	a	DET
ap-1412	108	7	⊂	⊂	X
ap-1412	108	8	â	â	X
ap-1412	108	9	=	=	SYM
ap-1412	108	10	(	(	PUNCT
ap-1412	108	11	â)∗	â)∗	PROPN
ap-1412	108	12	⊂	⊂	X
ap-1412	108	13	(	(	PUNCT
ap-1412	108	14	a)∗	a)∗	PROPN
ap-1412	108	15	⊂	⊂	PROPN
ap-1412	108	16	a∗.	a∗.	VERB
ap-1412	108	17	further	far	ADV
ap-1412	108	18	a∗∗	a∗∗	PUNCT
ap-1412	109	1	=	=	PUNCT
ap-1412	109	2	a	a	PRON
ap-1412	109	3	and	and	CCONJ
ap-1412	109	4	(	(	PUNCT
ap-1412	109	5	a)∗	a)∗	NOUN
ap-1412	109	6	=	=	SYM
ap-1412	109	7	a∗	a∗	PROPN
ap-1412	109	8	(	(	PUNCT
ap-1412	109	9	see	see	VERB
ap-1412	109	10	[	[	X
ap-1412	109	11	3	3	NUM
ap-1412	109	12	,	,	PUNCT
ap-1412	109	13	p.	p.	NOUN
ap-1412	109	14	96	96	NUM
ap-1412	109	15	]	]	PUNCT
ap-1412	109	16	)	)	PUNCT
ap-1412	109	17	.	.	PUNCT
ap-1412	110	1	(	(	PUNCT
ap-1412	110	2	ii	ii	NOUN
ap-1412	110	3	)	)	PUNCT
ap-1412	110	4	if	if	SCONJ
ap-1412	110	5	a	a	PRON
ap-1412	110	6	is	be	AUX
ap-1412	110	7	essentially	essentially	ADV
ap-1412	110	8	self	self	NOUN
ap-1412	110	9	-	-	PUNCT
ap-1412	110	10	adjoint	adjoint	NOUN
ap-1412	110	11	then	then	ADV
ap-1412	110	12	(	(	PUNCT
ap-1412	110	13	a)∗	a)∗	PROPN
ap-1412	110	14	=	=	PUNCT
ap-1412	110	15	a	a	PRON
ap-1412	110	16	implies	imply	VERB
ap-1412	110	17	â	â	PUNCT
ap-1412	111	1	=	=	NOUN
ap-1412	111	2	a.	a.	NOUN
ap-1412	111	3	conversely	conversely	ADV
ap-1412	111	4	,	,	PUNCT
ap-1412	111	5	if	if	SCONJ
ap-1412	111	6	a	a	PRON
ap-1412	111	7	=	=	X
ap-1412	111	8	â	â	ADP
ap-1412	111	9	then	then	ADV
ap-1412	111	10	a	a	PRON
ap-1412	111	11	=	=	X
ap-1412	111	12	â	â	X
ap-1412	111	13	=	=	PUNCT
ap-1412	111	14	(	(	PUNCT
ap-1412	111	15	â)∗	â)∗	PROPN
ap-1412	111	16	=	=	SYM
ap-1412	111	17	(	(	PUNCT
ap-1412	111	18	a)∗.	a)∗.	PROPN
ap-1412	111	19	(	(	PUNCT
ap-1412	111	20	iii	iii	NOUN
ap-1412	111	21	)	)	PUNCT
ap-1412	111	22	if	if	SCONJ
ap-1412	111	23	a	a	DET
ap-1412	111	24	∈	∈	PROPN
ap-1412	111	25	sp(h	sp(h	X
ap-1412	111	26	)	)	PUNCT
ap-1412	111	27	then	then	ADV
ap-1412	111	28	a	a	DET
ap-1412	111	29	=	=	X
ap-1412	111	30	a∗	a∗	NOUN
ap-1412	111	31	,	,	PUNCT
ap-1412	111	32	hence	hence	ADV
ap-1412	111	33	,	,	PUNCT
ap-1412	111	34	by	by	ADP
ap-1412	111	35	(	(	PUNCT
ap-1412	111	36	i	i	NOUN
ap-1412	111	37	)	)	PUNCT
ap-1412	111	38	,	,	PUNCT
ap-1412	111	39	a	a	DET
ap-1412	111	40	=	=	SYM
ap-1412	111	41	â	â	X
ap-1412	111	42	∈	∈	PROPN
ap-1412	111	43	v(h	v(h	NOUN
ap-1412	111	44	)	)	PUNCT
ap-1412	111	45	.	.	PUNCT
ap-1412	112	1	conversely	conversely	ADV
ap-1412	112	2	,	,	PUNCT
ap-1412	112	3	if	if	SCONJ
ap-1412	112	4	a	a	DET
ap-1412	112	5	∈	∈	PROPN
ap-1412	112	6	v(h	v(h	NOUN
ap-1412	112	7	)	)	PUNCT
ap-1412	112	8	then	then	ADV
ap-1412	112	9	a	a	PRON
ap-1412	112	10	is	be	AUX
ap-1412	112	11	self	self	NOUN
ap-1412	112	12	-	-	PUNCT
ap-1412	112	13	adjoint	adjoint	NOUN
ap-1412	112	14	,	,	PUNCT
ap-1412	112	15	hence	hence	ADV
ap-1412	112	16	a	a	DET
ap-1412	112	17	∈	∈	NOUN
ap-1412	112	18	sp(h	sp(h	NOUN
ap-1412	112	19	)	)	PUNCT
ap-1412	112	20	.	.	PUNCT
ap-1412	113	1	theorem	theorem	VERB
ap-1412	113	2	3.4	3.4	NUM
ap-1412	113	3	under	under	ADP
ap-1412	113	4	the	the	DET
ap-1412	113	5	assumption	assumption	NOUN
ap-1412	113	6	of	of	ADP
ap-1412	113	7	theorem	theorem	ADJ
ap-1412	113	8	3.2	3.2	NUM
ap-1412	113	9	let	let	VERB
ap-1412	113	10	sp(h	sp(h	X
ap-1412	113	11	)	)	PUNCT
ap-1412	114	1	=	=	SYM
ap-1412	114	2	{	{	PUNCT
ap-1412	114	3	a	a	DET
ap-1412	114	4	∈	∈	PROPN
ap-1412	114	5	v(h	v(h	NOUN
ap-1412	114	6	)	)	PUNCT
ap-1412	114	7	|	|	ADV
ap-1412	114	8	a	a	DET
ap-1412	114	9	=	=	NOUN
ap-1412	114	10	a∗	a∗	NOUN
ap-1412	114	11	}	}	PUNCT
ap-1412	114	12	and	and	CCONJ
ap-1412	114	13	let	let	VERB
ap-1412	114	14	⊕s	⊕s	NOUN
ap-1412	114	15	=	=	SYM
ap-1412	114	16	⊕/sp(h	⊕/sp(h	X
ap-1412	114	17	)	)	PUNCT
ap-1412	114	18	be	be	VERB
ap-1412	114	19	the	the	DET
ap-1412	114	20	restriction	restriction	NOUN
ap-1412	114	21	of	of	ADP
ap-1412	114	22	⊕-operation	⊕-operation	NOUN
ap-1412	114	23	defined	define	VERB
ap-1412	114	24	on	on	ADP
ap-1412	114	25	v(h	v(h	NOUN
ap-1412	114	26	)	)	PUNCT
ap-1412	114	27	to	to	ADP
ap-1412	114	28	the	the	DET
ap-1412	114	29	set	set	NOUN
ap-1412	114	30	sp(h	sp(h	NOUN
ap-1412	114	31	)	)	PUNCT
ap-1412	114	32	.	.	PUNCT
ap-1412	115	1	then	then	ADV
ap-1412	115	2	(	(	PUNCT
ap-1412	115	3	sp(h);⊕s	sp(h);⊕s	PROPN
ap-1412	115	4	,	,	PUNCT
ap-1412	115	5	0	0	NUM
ap-1412	115	6	)	)	PUNCT
ap-1412	115	7	is	be	AUX
ap-1412	115	8	a	a	DET
ap-1412	115	9	sub	sub	ADJ
ap-1412	115	10	-	-	ADJ
ap-1412	115	11	generalized	generalized	ADJ
ap-1412	115	12	effect	effect	NOUN
ap-1412	115	13	algebra	algebra	NOUN
ap-1412	115	14	of	of	ADP
ap-1412	115	15	(	(	PUNCT
ap-1412	115	16	v(h);⊕	v(h);⊕	NOUN
ap-1412	115	17	,	,	PUNCT
ap-1412	115	18	0	0	NUM
ap-1412	115	19	)	)	PUNCT
ap-1412	115	20	.	.	PUNCT
ap-1412	116	1	proof	proof	NOUN
ap-1412	116	2	.	.	PUNCT
ap-1412	117	1	we	we	PRON
ap-1412	117	2	have	have	VERB
ap-1412	117	3	to	to	PART
ap-1412	117	4	show	show	VERB
ap-1412	117	5	that	that	SCONJ
ap-1412	117	6	if	if	SCONJ
ap-1412	117	7	a	a	DET
ap-1412	117	8	,	,	PUNCT
ap-1412	117	9	b	b	NOUN
ap-1412	117	10	,	,	PUNCT
ap-1412	117	11	c	c	PROPN
ap-1412	117	12	∈	∈	PROPN
ap-1412	117	13	v(h	v(h	NOUN
ap-1412	117	14	)	)	PUNCT
ap-1412	117	15	with	with	ADP
ap-1412	117	16	a	a	DET
ap-1412	117	17	⊕	⊕	PROPN
ap-1412	117	18	b	b	PROPN
ap-1412	117	19	=	=	SYM
ap-1412	117	20	c	c	PROPN
ap-1412	117	21	and	and	CCONJ
ap-1412	117	22	out	out	ADP
ap-1412	117	23	of	of	ADP
ap-1412	117	24	a	a	DET
ap-1412	117	25	,	,	PUNCT
ap-1412	117	26	b	b	NOUN
ap-1412	117	27	,	,	PUNCT
ap-1412	117	28	c	c	NOUN
ap-1412	117	29	at	at	ADV
ap-1412	117	30	least	least	ADJ
ap-1412	117	31	two	two	NUM
ap-1412	117	32	are	be	AUX
ap-1412	117	33	in	in	ADP
ap-1412	117	34	sp(h	sp(h	NOUN
ap-1412	117	35	)	)	PUNCT
ap-1412	117	36	then	then	ADV
ap-1412	117	37	a	a	DET
ap-1412	117	38	,	,	PUNCT
ap-1412	117	39	b	b	NOUN
ap-1412	117	40	,	,	PUNCT
ap-1412	117	41	c	c	PROPN
ap-1412	117	42	∈	∈	PROPN
ap-1412	117	43	sp(h	sp(h	NOUN
ap-1412	117	44	)	)	PUNCT
ap-1412	117	45	.	.	PUNCT
ap-1412	118	1	(	(	PUNCT
ap-1412	118	2	i	i	NOUN
ap-1412	118	3	)	)	PUNCT
ap-1412	118	4	assume	assume	VERB
ap-1412	118	5	first	first	ADV
ap-1412	118	6	that	that	SCONJ
ap-1412	118	7	a	a	X
ap-1412	118	8	,	,	PUNCT
ap-1412	118	9	b	b	NOUN
ap-1412	118	10	∈	∈	PROPN
ap-1412	118	11	sp(h	sp(h	NOUN
ap-1412	118	12	)	)	PUNCT
ap-1412	118	13	.	.	PUNCT
ap-1412	119	1	if	if	SCONJ
ap-1412	119	2	a	a	DET
ap-1412	119	3	,	,	PUNCT
ap-1412	119	4	b	b	PROPN
ap-1412	119	5	are	be	AUX
ap-1412	119	6	bounded	bound	VERB
ap-1412	119	7	then	then	ADV
ap-1412	119	8	c	c	X
ap-1412	119	9	=	=	PUNCT
ap-1412	119	10	a	a	DET
ap-1412	119	11	⊕	⊕	PROPN
ap-1412	119	12	b	b	PROPN
ap-1412	119	13	is	be	AUX
ap-1412	119	14	again	again	ADV
ap-1412	119	15	bounded	bound	VERB
ap-1412	119	16	and	and	CCONJ
ap-1412	119	17	d(a	d(a	PROPN
ap-1412	119	18	)	)	PUNCT
ap-1412	119	19	=	=	SYM
ap-1412	120	1	d(b	d(b	X
ap-1412	120	2	)	)	PUNCT
ap-1412	120	3	=	=	SYM
ap-1412	120	4	d(c	d(c	PROPN
ap-1412	120	5	)	)	PUNCT
ap-1412	121	1	=	=	SYM
ap-1412	121	2	h	h	NOUN
ap-1412	121	3	,	,	PUNCT
ap-1412	121	4	hence	hence	ADV
ap-1412	121	5	c	c	NOUN
ap-1412	121	6	∈	∈	PROPN
ap-1412	121	7	sp(h	sp(h	NOUN
ap-1412	121	8	)	)	PUNCT
ap-1412	121	9	.	.	PUNCT
ap-1412	122	1	further	far	ADV
ap-1412	122	2	,	,	PUNCT
ap-1412	122	3	if	if	SCONJ
ap-1412	122	4	a	a	PRON
ap-1412	122	5	is	be	AUX
ap-1412	122	6	unbounded	unbounded	ADJ
ap-1412	122	7	and	and	CCONJ
ap-1412	122	8	b	b	NOUN
ap-1412	122	9	is	be	AUX
ap-1412	122	10	bounded	bound	VERB
ap-1412	122	11	then	then	ADV
ap-1412	122	12	c	c	X
ap-1412	122	13	=	=	PUNCT
ap-1412	123	1	a	a	DET
ap-1412	123	2	+	+	X
ap-1412	123	3	b|d(a	b|d(a	NOUN
ap-1412	123	4	)	)	PUNCT
ap-1412	123	5	and	and	CCONJ
ap-1412	123	6	d(c	d(c	PROPN
ap-1412	123	7	)	)	PUNCT
ap-1412	124	1	=	=	SYM
ap-1412	124	2	d(a	d(a	PROPN
ap-1412	124	3	)	)	PUNCT
ap-1412	124	4	.	.	PUNCT
ap-1412	125	1	moreover	moreover	ADV
ap-1412	125	2	,	,	PUNCT
ap-1412	125	3	a	a	DET
ap-1412	125	4	,	,	PUNCT
ap-1412	125	5	b	b	NOUN
ap-1412	125	6	∈	∈	PROPN
ap-1412	125	7	sp(h	sp(h	X
ap-1412	125	8	)	)	PUNCT
ap-1412	125	9	implies	imply	VERB
ap-1412	125	10	that	that	SCONJ
ap-1412	125	11	a	a	DET
ap-1412	125	12	=	=	NOUN
ap-1412	125	13	a∗	a∗	NOUN
ap-1412	125	14	,	,	PUNCT
ap-1412	125	15	hence	hence	ADV
ap-1412	125	16	d(a	d(a	PROPN
ap-1412	125	17	)	)	PUNCT
ap-1412	125	18	=	=	SYM
ap-1412	125	19	d(a∗	d(a∗	X
ap-1412	125	20	)	)	PUNCT
ap-1412	125	21	and	and	CCONJ
ap-1412	125	22	b	b	X
ap-1412	125	23	=	=	SYM
ap-1412	125	24	b∗	b∗	ADJ
ap-1412	125	25	⊂	⊂	X
ap-1412	125	26	(	(	PUNCT
ap-1412	125	27	b|d(a))∗	b|d(a))∗	PROPN
ap-1412	125	28	,	,	PUNCT
ap-1412	125	29	which	which	PRON
ap-1412	125	30	gives	give	VERB
ap-1412	125	31	b	b	PROPN
ap-1412	125	32	=	=	SYM
ap-1412	125	33	(	(	PUNCT
ap-1412	125	34	b|d(a))∗	b|d(a))∗	PROPN
ap-1412	125	35	on	on	ADP
ap-1412	125	36	h.	h.	PROPN
ap-1412	125	37	it	it	PRON
ap-1412	125	38	follows	follow	VERB
ap-1412	125	39	,	,	PUNCT
ap-1412	125	40	as	as	ADP
ap-1412	125	41	b|d(a	b|d(a	PROPN
ap-1412	125	42	)	)	PUNCT
ap-1412	125	43	is	be	AUX
ap-1412	125	44	bounded	bound	VERB
ap-1412	125	45	,	,	PUNCT
ap-1412	125	46	that	that	SCONJ
ap-1412	125	47	(	(	PUNCT
ap-1412	125	48	a⊕b)∗	a⊕b)∗	PUNCT
ap-1412	125	49	=	=	SYM
ap-1412	125	50	(	(	PUNCT
ap-1412	125	51	a+b|d(a))∗	a+b|d(a))∗	NOUN
ap-1412	125	52	=	=	NOUN
ap-1412	125	53	a∗+(b|d(a))∗	a∗+(b|d(a))∗	NOUN
ap-1412	125	54	=	=	SYM
ap-1412	125	55	a∗	a∗	PROPN
ap-1412	125	56	+	+	CCONJ
ap-1412	125	57	b	b	NOUN
ap-1412	125	58	=	=	SYM
ap-1412	125	59	a∗	a∗	NOUN
ap-1412	125	60	+	+	CCONJ
ap-1412	125	61	b|d(a∗	b|d(a∗	NOUN
ap-1412	125	62	)	)	PUNCT
ap-1412	125	63	=	=	SYM
ap-1412	125	64	a	a	DET
ap-1412	125	65	+	+	NUM
ap-1412	125	66	b|d(a	b|d(a	NOUN
ap-1412	125	67	)	)	PUNCT
ap-1412	125	68	=	=	PUNCT
ap-1412	125	69	a	a	DET
ap-1412	125	70	⊕	⊕	PROPN
ap-1412	125	71	b.	b.	PROPN
ap-1412	125	72	again	again	ADV
ap-1412	125	73	c	c	PROPN
ap-1412	125	74	∈	∈	PROPN
ap-1412	125	75	sp(h	sp(h	NOUN
ap-1412	125	76	)	)	PUNCT
ap-1412	125	77	.	.	PUNCT
ap-1412	126	1	(	(	PUNCT
ap-1412	126	2	ii	ii	NOUN
ap-1412	126	3	)	)	PUNCT
ap-1412	126	4	assume	assume	VERB
ap-1412	126	5	now	now	ADV
ap-1412	126	6	that	that	SCONJ
ap-1412	126	7	a	a	X
ap-1412	126	8	,	,	PUNCT
ap-1412	126	9	c	c	PROPN
ap-1412	126	10	∈	∈	PROPN
ap-1412	126	11	sp(h	sp(h	NOUN
ap-1412	126	12	)	)	PUNCT
ap-1412	126	13	.	.	PUNCT
ap-1412	127	1	then	then	ADV
ap-1412	127	2	if	if	SCONJ
ap-1412	127	3	c	c	PROPN
ap-1412	127	4	is	be	AUX
ap-1412	127	5	bounded	bound	VERB
ap-1412	127	6	then	then	ADV
ap-1412	127	7	d(c	d(c	PROPN
ap-1412	127	8	)	)	PUNCT
ap-1412	128	1	=	=	SYM
ap-1412	128	2	h	h	NOUN
ap-1412	128	3	and	and	CCONJ
ap-1412	128	4	then	then	ADV
ap-1412	128	5	a	a	DET
ap-1412	128	6	,	,	PUNCT
ap-1412	128	7	b	b	PROPN
ap-1412	128	8	are	be	AUX
ap-1412	128	9	bounded	bound	VERB
ap-1412	128	10	,	,	PUNCT
ap-1412	128	11	hence	hence	ADV
ap-1412	128	12	a	a	PRON
ap-1412	128	13	,	,	PUNCT
ap-1412	128	14	b	b	X
ap-1412	128	15	∈	∈	PROPN
ap-1412	128	16	sp(h	sp(h	NOUN
ap-1412	128	17	)	)	PUNCT
ap-1412	128	18	.	.	PUNCT
ap-1412	129	1	if	if	SCONJ
ap-1412	129	2	c	c	PROPN
ap-1412	129	3	and	and	CCONJ
ap-1412	129	4	a	a	PRON
ap-1412	129	5	are	be	AUX
ap-1412	129	6	unbounded	unbounded	ADJ
ap-1412	129	7	then	then	ADV
ap-1412	129	8	b	b	PROPN
ap-1412	129	9	is	be	AUX
ap-1412	129	10	bounded	bound	VERB
ap-1412	129	11	(	(	PUNCT
ap-1412	129	12	since	since	SCONJ
ap-1412	129	13	otherwise	otherwise	ADV
ap-1412	129	14	a	a	DET
ap-1412	129	15	⊕	⊕	PROPN
ap-1412	129	16	b	b	PROPN
ap-1412	129	17	is	be	AUX
ap-1412	129	18	not	not	PART
ap-1412	129	19	defined	define	VERB
ap-1412	129	20	)	)	PUNCT
ap-1412	129	21	and	and	CCONJ
ap-1412	129	22	again	again	ADV
ap-1412	129	23	b	b	PROPN
ap-1412	129	24	∈	∈	PROPN
ap-1412	129	25	sp(h	sp(h	NOUN
ap-1412	129	26	)	)	PUNCT
ap-1412	129	27	.	.	PUNCT
ap-1412	130	1	finally	finally	ADV
ap-1412	130	2	,	,	PUNCT
ap-1412	130	3	if	if	SCONJ
ap-1412	130	4	c	c	PROPN
ap-1412	130	5	is	be	AUX
ap-1412	130	6	unbounded	unbounded	ADJ
ap-1412	130	7	and	and	CCONJ
ap-1412	130	8	a	a	PRON
ap-1412	130	9	is	be	AUX
ap-1412	130	10	bounded	bound	VERB
ap-1412	130	11	then	then	ADV
ap-1412	130	12	b	b	NOUN
ap-1412	130	13	is	be	AUX
ap-1412	130	14	unbounded	unbounded	ADJ
ap-1412	130	15	and	and	CCONJ
ap-1412	130	16	c	c	NOUN
ap-1412	130	17	=	=	SYM
ap-1412	130	18	a|d(b)+b	a|d(b)+b	PROPN
ap-1412	130	19	.	.	PUNCT
ap-1412	131	1	it	it	PRON
ap-1412	131	2	follows	follow	VERB
ap-1412	131	3	that	that	SCONJ
ap-1412	131	4	d(c	d(c	PROPN
ap-1412	131	5	)	)	PUNCT
ap-1412	131	6	=	=	PUNCT
ap-1412	131	7	d(b	d(b	PROPN
ap-1412	131	8	)	)	PUNCT
ap-1412	131	9	.	.	PUNCT
ap-1412	132	1	moreover	moreover	ADV
ap-1412	132	2	,	,	PUNCT
ap-1412	132	3	c∗	c∗	PROPN
ap-1412	132	4	=	=	SYM
ap-1412	132	5	(	(	PUNCT
ap-1412	132	6	a|d(b))∗	a|d(b))∗	NOUN
ap-1412	132	7	+	+	CCONJ
ap-1412	132	8	b∗	b∗	ADJ
ap-1412	132	9	=	=	PUNCT
ap-1412	132	10	a	a	PRON
ap-1412	132	11	+	+	X
ap-1412	132	12	b∗	b∗	ADJ
ap-1412	132	13	,	,	PUNCT
ap-1412	132	14	hence	hence	ADV
ap-1412	132	15	d(c∗	d(c∗	NOUN
ap-1412	132	16	)	)	PUNCT
ap-1412	132	17	=	=	SYM
ap-1412	132	18	d(b∗	d(b∗	NOUN
ap-1412	132	19	)	)	PUNCT
ap-1412	132	20	.	.	PUNCT
ap-1412	133	1	now	now	ADV
ap-1412	133	2	,	,	PUNCT
ap-1412	133	3	the	the	DET
ap-1412	133	4	assumption	assumption	NOUN
ap-1412	133	5	that	that	SCONJ
ap-1412	133	6	c	c	PROPN
ap-1412	133	7	is	be	AUX
ap-1412	133	8	self	self	NOUN
ap-1412	133	9	-	-	PUNCT
ap-1412	133	10	adjoint	adjoint	NOUN
ap-1412	133	11	implies	imply	VERB
ap-1412	133	12	d(c	d(c	PROPN
ap-1412	133	13	)	)	PUNCT
ap-1412	134	1	=	=	SYM
ap-1412	134	2	d(c∗	d(c∗	NOUN
ap-1412	134	3	)	)	PUNCT
ap-1412	134	4	which	which	PRON
ap-1412	134	5	gives	give	VERB
ap-1412	134	6	d(b∗	d(b∗	NOUN
ap-1412	134	7	)	)	PUNCT
ap-1412	134	8	=	=	PUNCT
ap-1412	134	9	d(b	d(b	PROPN
ap-1412	134	10	)	)	PUNCT
ap-1412	134	11	,	,	PUNCT
ap-1412	134	12	hence	hence	ADV
ap-1412	134	13	b	b	X
ap-1412	134	14	∈	∈	PROPN
ap-1412	134	15	sp(h	sp(h	NOUN
ap-1412	134	16	)	)	PUNCT
ap-1412	134	17	.	.	PUNCT
ap-1412	135	1	in	in	ADP
ap-1412	135	2	theorem	theorem	NOUN
ap-1412	135	3	3.4	3.4	NUM
ap-1412	135	4	we	we	PRON
ap-1412	135	5	may	may	AUX
ap-1412	135	6	substitute	substitute	VERB
ap-1412	135	7	sp(h	sp(h	NOUN
ap-1412	135	8	)	)	PUNCT
ap-1412	135	9	by	by	ADP
ap-1412	135	10	f(h	f(h	PROPN
ap-1412	135	11	)	)	PUNCT
ap-1412	135	12	.	.	PUNCT
ap-1412	136	1	hence	hence	ADV
ap-1412	136	2	f(h	f(h	PROPN
ap-1412	136	3	)	)	PUNCT
ap-1412	136	4	is	be	AUX
ap-1412	136	5	a	a	DET
ap-1412	136	6	generalized	generalized	ADJ
ap-1412	136	7	effect	effect	NOUN
ap-1412	136	8	algebra	algebra	NOUN
ap-1412	136	9	,	,	PUNCT
ap-1412	136	10	more	more	ADV
ap-1412	136	11	precisely	precisely	ADV
ap-1412	136	12	:	:	PUNCT
ap-1412	136	13	corollary	corollary	ADJ
ap-1412	136	14	3.5	3.5	NUM
ap-1412	136	15	let	let	VERB
ap-1412	136	16	h	h	NOUN
ap-1412	136	17	be	be	AUX
ap-1412	136	18	an	an	DET
ap-1412	136	19	infinite	infinite	ADJ
ap-1412	136	20	-	-	PUNCT
ap-1412	136	21	dimensional	dimensional	ADJ
ap-1412	136	22	complex	complex	ADJ
ap-1412	136	23	hilbert	hilbert	NOUN
ap-1412	136	24	space	space	NOUN
ap-1412	136	25	.	.	PUNCT
ap-1412	137	1	let	let	VERB
ap-1412	137	2	f(h	f(h	PROPN
ap-1412	137	3	)	)	PUNCT
ap-1412	137	4	be	be	AUX
ap-1412	137	5	the	the	DET
ap-1412	137	6	set	set	NOUN
ap-1412	137	7	of	of	ADP
ap-1412	137	8	all	all	DET
ap-1412	137	9	friedrichs	friedrich	NOUN
ap-1412	137	10	positive	positive	ADJ
ap-1412	137	11	self	self	NOUN
ap-1412	137	12	-	-	PUNCT
ap-1412	137	13	adjoint	adjoint	NOUN
ap-1412	137	14	extensions	extension	NOUN
ap-1412	137	15	of	of	ADP
ap-1412	137	16	all	all	DET
ap-1412	137	17	positive	positive	ADJ
ap-1412	137	18	densely	densely	ADV
ap-1412	137	19	defined	define	VERB
ap-1412	137	20	linear	linear	NOUN
ap-1412	137	21	operators	operator	NOUN
ap-1412	137	22	in	in	ADP
ap-1412	137	23	h	h	NOUN
ap-1412	137	24	with	with	ADP
ap-1412	137	25	d(a	d(a	PROPN
ap-1412	137	26	)	)	PUNCT
ap-1412	138	1	=	=	SYM
ap-1412	138	2	h	h	NOUN
ap-1412	138	3	if	if	SCONJ
ap-1412	138	4	a	a	PRON
ap-1412	138	5	is	be	AUX
ap-1412	138	6	bounded	bound	VERB
ap-1412	138	7	.	.	PUNCT
ap-1412	139	1	let	let	VERB
ap-1412	139	2	⊕	⊕	PROPN
ap-1412	139	3	be	be	AUX
ap-1412	139	4	a	a	DET
ap-1412	139	5	partial	partial	ADJ
ap-1412	139	6	binary	binary	ADJ
ap-1412	139	7	operation	operation	NOUN
ap-1412	139	8	defined	define	VERB
ap-1412	139	9	for	for	ADP
ap-1412	139	10	a	a	DET
ap-1412	139	11	,	,	PUNCT
ap-1412	139	12	b	b	PROPN
ap-1412	139	13	∈	∈	PROPN
ap-1412	139	14	f(h	f(h	PROPN
ap-1412	139	15	)	)	PUNCT
ap-1412	139	16	iff	iff	VERB
ap-1412	139	17	out	out	ADP
ap-1412	139	18	80	80	NUM
ap-1412	139	19	acta	acta	PROPN
ap-1412	139	20	polytechnica	polytechnica	PROPN
ap-1412	139	21	vol	vol	NOUN
ap-1412	139	22	.	.	PUNCT
ap-1412	140	1	51	51	NUM
ap-1412	140	2	no	no	INTJ
ap-1412	140	3	.	.	PUNCT
ap-1412	141	1	4/2011	4/2011	NUM
ap-1412	141	2	of	of	ADP
ap-1412	141	3	operators	operator	NOUN
ap-1412	141	4	a	a	DET
ap-1412	141	5	,	,	PUNCT
ap-1412	141	6	b	b	NOUN
ap-1412	141	7	at	at	ADV
ap-1412	141	8	least	least	ADJ
ap-1412	141	9	one	one	NUM
ap-1412	141	10	is	be	AUX
ap-1412	141	11	bounded	bound	VERB
ap-1412	142	1	and	and	CCONJ
ap-1412	142	2	then	then	ADV
ap-1412	142	3	a	a	DET
ap-1412	142	4	⊕	⊕	PROPN
ap-1412	142	5	b	b	X
ap-1412	142	6	=	=	PUNCT
ap-1412	143	1	a	a	DET
ap-1412	143	2	+	+	NUM
ap-1412	143	3	b	b	NOUN
ap-1412	143	4	is	be	AUX
ap-1412	143	5	the	the	DET
ap-1412	143	6	usual	usual	ADJ
ap-1412	143	7	sum	sum	NOUN
ap-1412	143	8	of	of	ADP
ap-1412	143	9	operators	operator	NOUN
ap-1412	143	10	in	in	ADP
ap-1412	143	11	h.	h.	PROPN
ap-1412	143	12	then	then	ADV
ap-1412	143	13	(	(	PUNCT
ap-1412	143	14	f(h);⊕	f(h);⊕	NOUN
ap-1412	143	15	,	,	PUNCT
ap-1412	143	16	0	0	NUM
ap-1412	143	17	)	)	PUNCT
ap-1412	143	18	is	be	AUX
ap-1412	143	19	a	a	DET
ap-1412	143	20	generalized	generalized	ADJ
ap-1412	143	21	effect	effect	NOUN
ap-1412	143	22	algebra	algebra	NOUN
ap-1412	143	23	.	.	PUNCT
ap-1412	144	1	assume	assume	VERB
ap-1412	144	2	that	that	SCONJ
ap-1412	144	3	(	(	PUNCT
ap-1412	144	4	e;⊕	e;⊕	ADJ
ap-1412	144	5	,	,	PUNCT
ap-1412	144	6	0	0	NUM
ap-1412	144	7	)	)	PUNCT
ap-1412	144	8	is	be	AUX
ap-1412	144	9	a	a	DET
ap-1412	144	10	generalized	generalized	ADJ
ap-1412	144	11	effect	effect	NOUN
ap-1412	144	12	algebra	algebra	NOUN
ap-1412	144	13	.	.	PUNCT
ap-1412	145	1	then	then	ADV
ap-1412	145	2	(	(	PUNCT
ap-1412	145	3	see	see	VERB
ap-1412	145	4	,	,	PUNCT
ap-1412	145	5	e.g.	e.g.	ADV
ap-1412	145	6	,	,	PUNCT
ap-1412	145	7	[	[	X
ap-1412	145	8	11	11	NUM
ap-1412	145	9	]	]	PUNCT
ap-1412	145	10	)	)	PUNCT
ap-1412	145	11	for	for	ADP
ap-1412	145	12	any	any	DET
ap-1412	145	13	fixed	fix	VERB
ap-1412	145	14	q	q	NOUN
ap-1412	145	15	∈	∈	PROPN
ap-1412	145	16	e	e	NOUN
ap-1412	145	17	,	,	PUNCT
ap-1412	145	18	q	q	PROPN
ap-1412	145	19	�	�	PROPN
ap-1412	145	20	=	=	NOUN
ap-1412	145	21	0	0	NUM
ap-1412	146	1	the	the	DET
ap-1412	146	2	interval	interval	NOUN
ap-1412	146	3	[	[	X
ap-1412	146	4	0	0	NUM
ap-1412	146	5	,	,	PUNCT
ap-1412	146	6	q]e	q]e	ADJ
ap-1412	146	7	=	=	SYM
ap-1412	146	8	{	{	PUNCT
ap-1412	146	9	x	x	SYM
ap-1412	146	10	∈	∈	NOUN
ap-1412	146	11	e	e	NOUN
ap-1412	146	12	|	|	ADV
ap-1412	146	13	there	there	PRON
ap-1412	146	14	exists	exist	VERB
ap-1412	146	15	y	y	PROPN
ap-1412	146	16	∈	∈	PROPN
ap-1412	146	17	e	e	X
ap-1412	146	18	with	with	ADP
ap-1412	146	19	x⊕y	x⊕y	PROPN
ap-1412	146	20	=	=	PUNCT
ap-1412	146	21	q	q	X
ap-1412	146	22	}	}	PUNCT
ap-1412	146	23	is	be	AUX
ap-1412	146	24	an	an	DET
ap-1412	146	25	effect	effect	NOUN
ap-1412	146	26	algebra	algebra	NOUN
ap-1412	146	27	(	(	PUNCT
ap-1412	146	28	[	[	X
ap-1412	146	29	0	0	NUM
ap-1412	146	30	,	,	PUNCT
ap-1412	146	31	q]e	q]e	ADJ
ap-1412	146	32	;	;	PUNCT
ap-1412	146	33	⊕q	⊕q	PROPN
ap-1412	146	34	,	,	PUNCT
ap-1412	146	35	0	0	NUM
ap-1412	146	36	,	,	PUNCT
ap-1412	146	37	q	q	NOUN
ap-1412	146	38	)	)	PUNCT
ap-1412	146	39	with	with	ADP
ap-1412	146	40	unit	unit	NOUN
ap-1412	146	41	q	q	PROPN
ap-1412	146	42	and	and	CCONJ
ap-1412	146	43	with	with	SCONJ
ap-1412	146	44	the	the	DET
ap-1412	146	45	partial	partial	ADJ
ap-1412	146	46	operation	operation	NOUN
ap-1412	146	47	⊕q	⊕q	NOUN
ap-1412	146	48	defined	define	VERB
ap-1412	146	49	for	for	ADP
ap-1412	146	50	x	x	X
ap-1412	146	51	,	,	PUNCT
ap-1412	146	52	y	y	PROPN
ap-1412	146	53	∈	∈	PROPN
ap-1412	147	1	[	[	X
ap-1412	147	2	0	0	NUM
ap-1412	147	3	,	,	PUNCT
ap-1412	147	4	q]e	q]e	VERB
ap-1412	147	5	by	by	ADP
ap-1412	147	6	x	x	SYM
ap-1412	147	7	⊕q	⊕q	PROPN
ap-1412	147	8	y	y	PROPN
ap-1412	147	9	exists	exist	VERB
ap-1412	147	10	and	and	CCONJ
ap-1412	147	11	x	x	ADP
ap-1412	147	12	⊕q	⊕q	NOUN
ap-1412	147	13	y	y	PROPN
ap-1412	147	14	=	=	PUNCT
ap-1412	147	15	x	x	SYM
ap-1412	147	16	⊕	⊕	PROPN
ap-1412	147	17	y	y	PROPN
ap-1412	147	18	iff	iff	PROPN
ap-1412	147	19	x	x	PROPN
ap-1412	147	20	⊕	⊕	PROPN
ap-1412	147	21	y	y	PROPN
ap-1412	147	22	∈	∈	PROPN
ap-1412	148	1	[	[	X
ap-1412	148	2	0	0	NUM
ap-1412	148	3	,	,	PUNCT
ap-1412	148	4	q]e	q]e	ADJ
ap-1412	148	5	exists	exist	NOUN
ap-1412	148	6	in	in	ADP
ap-1412	148	7	e	e	NOUN
ap-1412	148	8	.	.	PUNCT
ap-1412	149	1	we	we	PRON
ap-1412	149	2	have	have	AUX
ap-1412	149	3	shown	show	VERB
ap-1412	149	4	that	that	SCONJ
ap-1412	149	5	v(h	v(h	NOUN
ap-1412	149	6	)	)	PUNCT
ap-1412	149	7	and	and	CCONJ
ap-1412	149	8	sp(h	sp(h	NOUN
ap-1412	149	9	)	)	PUNCT
ap-1412	149	10	are	be	AUX
ap-1412	149	11	generalized	generalized	ADJ
ap-1412	149	12	effect	effect	NOUN
ap-1412	149	13	algebras	algebra	NOUN
ap-1412	149	14	under	under	ADP
ap-1412	149	15	the	the	DET
ap-1412	149	16	partial	partial	ADJ
ap-1412	149	17	operations	operation	NOUN
ap-1412	149	18	⊕	⊕	PROPN
ap-1412	149	19	and	and	CCONJ
ap-1412	149	20	⊕s	⊕s	NOUN
ap-1412	149	21	,	,	PUNCT
ap-1412	149	22	respectively	respectively	ADV
ap-1412	149	23	.	.	PUNCT
ap-1412	150	1	moreover	moreover	ADV
ap-1412	150	2	,	,	PUNCT
ap-1412	150	3	a⊕b	a⊕b	PROPN
ap-1412	150	4	(	(	PUNCT
ap-1412	150	5	for	for	ADP
ap-1412	150	6	a	a	DET
ap-1412	150	7	,	,	PUNCT
ap-1412	150	8	b	b	PROPN
ap-1412	150	9	∈	∈	PROPN
ap-1412	150	10	v(h	v(h	NOUN
ap-1412	150	11	)	)	PUNCT
ap-1412	150	12	)	)	PUNCT
ap-1412	150	13	and	and	CCONJ
ap-1412	150	14	a	a	DET
ap-1412	150	15	⊕s	⊕s	NOUN
ap-1412	150	16	b	b	PROPN
ap-1412	150	17	(	(	PUNCT
ap-1412	150	18	for	for	ADP
ap-1412	150	19	a	a	DET
ap-1412	150	20	,	,	PUNCT
ap-1412	150	21	b	b	PROPN
ap-1412	150	22	∈	∈	PROPN
ap-1412	150	23	sp(h	sp(h	X
ap-1412	150	24	)	)	PUNCT
ap-1412	150	25	)	)	PUNCT
ap-1412	150	26	coincide	coincide	NOUN
ap-1412	150	27	with	with	ADP
ap-1412	150	28	the	the	DET
ap-1412	150	29	usual	usual	ADJ
ap-1412	150	30	sum	sum	NOUN
ap-1412	150	31	of	of	ADP
ap-1412	150	32	operators	operator	NOUN
ap-1412	150	33	a	a	PRON
ap-1412	150	34	,	,	PUNCT
ap-1412	150	35	b	b	PROPN
ap-1412	150	36	when	when	SCONJ
ap-1412	150	37	at	at	ADV
ap-1412	150	38	least	least	ADV
ap-1412	150	39	one	one	NUM
ap-1412	150	40	of	of	ADP
ap-1412	150	41	them	they	PRON
ap-1412	150	42	is	be	AUX
ap-1412	150	43	bounded	bound	VERB
ap-1412	150	44	.	.	PUNCT
ap-1412	151	1	if	if	SCONJ
ap-1412	151	2	both	both	PRON
ap-1412	151	3	a	a	DET
ap-1412	151	4	,	,	PUNCT
ap-1412	151	5	b	b	NOUN
ap-1412	151	6	are	be	AUX
ap-1412	151	7	unbounded	unbounded	ADJ
ap-1412	151	8	then	then	ADV
ap-1412	151	9	a⊕b	a⊕b	ADJ
ap-1412	151	10	,	,	PUNCT
ap-1412	151	11	a⊕sb	a⊕sb	PROPN
ap-1412	151	12	,	,	PUNCT
ap-1412	151	13	respectively	respectively	ADV
ap-1412	151	14	,	,	PUNCT
ap-1412	151	15	are	be	AUX
ap-1412	151	16	not	not	PART
ap-1412	151	17	defined	define	VERB
ap-1412	151	18	.	.	PUNCT
ap-1412	152	1	since	since	SCONJ
ap-1412	152	2	for	for	ADP
ap-1412	152	3	any	any	DET
ap-1412	152	4	fixed	fix	VERB
ap-1412	152	5	q	q	NOUN
ap-1412	152	6	∈	∈	PROPN
ap-1412	152	7	sp(h	sp(h	NOUN
ap-1412	152	8	)	)	PUNCT
ap-1412	152	9	,	,	PUNCT
ap-1412	152	10	q	q	PROPN
ap-1412	152	11	�	�	PROPN
ap-1412	152	12	=	=	SYM
ap-1412	152	13	0	0	NUM
ap-1412	152	14	,	,	PUNCT
ap-1412	152	15	it	it	PRON
ap-1412	152	16	holds	hold	VERB
ap-1412	152	17	[	[	X
ap-1412	152	18	0	0	NUM
ap-1412	152	19	,	,	PUNCT
ap-1412	152	20	q]sp(h	q]sp(h	ADV
ap-1412	152	21	)	)	PUNCT
ap-1412	152	22	=	=	PUNCT
ap-1412	153	1	[	[	X
ap-1412	153	2	0	0	NUM
ap-1412	153	3	,	,	PUNCT
ap-1412	153	4	q]v(h	q]v(h	NOUN
ap-1412	153	5	)	)	PUNCT
ap-1412	153	6	∩	∩	NOUN
ap-1412	153	7	sp(h	sp(h	NOUN
ap-1412	153	8	)	)	PUNCT
ap-1412	153	9	,	,	PUNCT
ap-1412	153	10	we	we	PRON
ap-1412	153	11	obtain	obtain	VERB
ap-1412	153	12	the	the	DET
ap-1412	153	13	following	following	ADJ
ap-1412	153	14	effect	effect	NOUN
ap-1412	153	15	algebras	algebra	NOUN
ap-1412	153	16	of	of	ADP
ap-1412	153	17	positive	positive	ADJ
ap-1412	153	18	self	self	NOUN
ap-1412	153	19	-	-	PUNCT
ap-1412	153	20	adjoint	adjoint	NOUN
ap-1412	153	21	operators	operator	NOUN
ap-1412	153	22	:	:	PUNCT
ap-1412	153	23	theorem	theorem	VERB
ap-1412	153	24	3.6	3.6	NUM
ap-1412	153	25	let	let	VERB
ap-1412	153	26	q	q	PROPN
ap-1412	153	27	∈	∈	PROPN
ap-1412	153	28	sp(h	sp(h	NOUN
ap-1412	153	29	)	)	PUNCT
ap-1412	153	30	,	,	PUNCT
ap-1412	153	31	q	q	PROPN
ap-1412	153	32	�	�	PROPN
ap-1412	153	33	=	=	SYM
ap-1412	153	34	0	0	NUM
ap-1412	153	35	be	be	AUX
ap-1412	153	36	fixed	fix	VERB
ap-1412	153	37	.	.	PUNCT
ap-1412	154	1	then	then	ADV
ap-1412	154	2	(	(	PUNCT
ap-1412	154	3	[	[	X
ap-1412	154	4	0	0	NUM
ap-1412	154	5	,	,	PUNCT
ap-1412	154	6	q]sp(h);⊕q	q]sp(h);⊕q	NOUN
ap-1412	154	7	,	,	PUNCT
ap-1412	154	8	0	0	NUM
ap-1412	154	9	,	,	PUNCT
ap-1412	154	10	q	q	X
ap-1412	154	11	)	)	PUNCT
ap-1412	154	12	is	be	AUX
ap-1412	154	13	an	an	DET
ap-1412	154	14	effect	effect	NOUN
ap-1412	154	15	algebra	algebra	NOUN
ap-1412	154	16	(	(	PUNCT
ap-1412	154	17	with	with	ADP
ap-1412	154	18	unit	unit	NOUN
ap-1412	154	19	q	q	NOUN
ap-1412	154	20	)	)	PUNCT
ap-1412	154	21	of	of	ADP
ap-1412	154	22	positive	positive	ADJ
ap-1412	154	23	self	self	NOUN
ap-1412	154	24	-	-	PUNCT
ap-1412	154	25	adjoint	adjoint	NOUN
ap-1412	154	26	operators	operator	NOUN
ap-1412	154	27	densely	densely	ADV
ap-1412	154	28	defined	define	VERB
ap-1412	154	29	in	in	ADP
ap-1412	154	30	h	h	NOUN
ap-1412	154	31	under	under	ADP
ap-1412	154	32	the	the	DET
ap-1412	154	33	⊕q	⊕q	NOUN
ap-1412	154	34	defined	define	VERB
ap-1412	154	35	for	for	ADP
ap-1412	154	36	a	a	DET
ap-1412	154	37	,	,	PUNCT
ap-1412	154	38	b	b	PROPN
ap-1412	154	39	∈	∈	PROPN
ap-1412	155	1	[	[	X
ap-1412	155	2	0	0	NUM
ap-1412	155	3	,	,	PUNCT
ap-1412	155	4	q]sp(h	q]sp(h	ADV
ap-1412	155	5	)	)	PUNCT
ap-1412	155	6	by	by	ADP
ap-1412	155	7	:	:	PUNCT
ap-1412	155	8	a	a	DET
ap-1412	155	9	⊕q	⊕q	NOUN
ap-1412	155	10	b	b	PROPN
ap-1412	155	11	exists	exist	NOUN
ap-1412	155	12	and	and	CCONJ
ap-1412	155	13	a	a	DET
ap-1412	155	14	⊕q	⊕q	NOUN
ap-1412	155	15	b	b	X
ap-1412	155	16	=	=	PUNCT
ap-1412	155	17	a	a	DET
ap-1412	155	18	+	+	X
ap-1412	155	19	b	b	NOUN
ap-1412	155	20	(	(	PUNCT
ap-1412	155	21	the	the	DET
ap-1412	155	22	usual	usual	ADJ
ap-1412	155	23	sum	sum	NOUN
ap-1412	155	24	of	of	ADP
ap-1412	155	25	a	a	DET
ap-1412	155	26	,	,	PUNCT
ap-1412	155	27	b	b	NOUN
ap-1412	155	28	in	in	ADP
ap-1412	155	29	h	h	NOUN
ap-1412	155	30	)	)	PUNCT
ap-1412	155	31	iff	iff	NOUN
ap-1412	155	32	at	at	ADV
ap-1412	155	33	least	least	ADV
ap-1412	155	34	one	one	NUM
ap-1412	155	35	out	out	ADP
ap-1412	155	36	of	of	ADP
ap-1412	155	37	operators	operator	NOUN
ap-1412	155	38	a	a	PRON
ap-1412	155	39	,	,	PUNCT
ap-1412	155	40	b	b	PROPN
ap-1412	155	41	is	be	AUX
ap-1412	155	42	bounded	bound	VERB
ap-1412	155	43	and	and	CCONJ
ap-1412	155	44	a	a	DET
ap-1412	155	45	+	+	NOUN
ap-1412	155	46	b	b	NOUN
ap-1412	155	47	∈	∈	NOUN
ap-1412	156	1	[	[	X
ap-1412	156	2	0	0	NUM
ap-1412	156	3	,	,	PUNCT
ap-1412	156	4	q]v(h	q]v(h	NOUN
ap-1412	156	5	)	)	PUNCT
ap-1412	156	6	.	.	PUNCT
ap-1412	157	1	note	note	VERB
ap-1412	157	2	that	that	SCONJ
ap-1412	157	3	if	if	SCONJ
ap-1412	157	4	we	we	PRON
ap-1412	157	5	substitute	substitute	VERB
ap-1412	157	6	sp(h	sp(h	NOUN
ap-1412	157	7	)	)	PUNCT
ap-1412	157	8	in	in	ADP
ap-1412	157	9	the	the	DET
ap-1412	157	10	preceding	precede	VERB
ap-1412	157	11	theorem	theorem	NOUN
ap-1412	157	12	by	by	ADP
ap-1412	157	13	v(h	v(h	NOUN
ap-1412	157	14	)	)	PUNCT
ap-1412	157	15	then	then	ADV
ap-1412	157	16	for	for	ADP
ap-1412	157	17	every	every	DET
ap-1412	157	18	fixed	fix	VERB
ap-1412	157	19	q	q	PROPN
ap-1412	157	20	∈	∈	PROPN
ap-1412	157	21	v(h	v(h	NOUN
ap-1412	157	22	)	)	PUNCT
ap-1412	157	23	we	we	PRON
ap-1412	157	24	have	have	VERB
ap-1412	157	25	[	[	X
ap-1412	157	26	0	0	NUM
ap-1412	157	27	,	,	PUNCT
ap-1412	157	28	q]v(h	q]v(h	NOUN
ap-1412	157	29	)	)	PUNCT
ap-1412	157	30	=	=	PRON
ap-1412	157	31	{	{	PUNCT
ap-1412	157	32	a	a	DET
ap-1412	157	33	∈	∈	PROPN
ap-1412	157	34	v(h	v(h	NOUN
ap-1412	157	35	)	)	PUNCT
ap-1412	158	1	|	|	ADV
ap-1412	158	2	there	there	PRON
ap-1412	158	3	exists	exist	VERB
ap-1412	158	4	c	c	PROPN
ap-1412	158	5	∈	∈	PROPN
ap-1412	158	6	v(h	v(h	NOUN
ap-1412	158	7	)	)	PUNCT
ap-1412	158	8	such	such	ADJ
ap-1412	158	9	that	that	PRON
ap-1412	158	10	out	out	ADP
ap-1412	158	11	of	of	ADP
ap-1412	158	12	a	a	PRON
ap-1412	158	13	,	,	PUNCT
ap-1412	158	14	c	c	AUX
ap-1412	158	15	at	at	ADV
ap-1412	158	16	least	least	ADJ
ap-1412	158	17	one	one	NUM
ap-1412	158	18	is	be	AUX
ap-1412	158	19	bounded	bound	VERB
ap-1412	158	20	and	and	CCONJ
ap-1412	158	21	a	a	DET
ap-1412	158	22	+	+	NOUN
ap-1412	158	23	c	c	NOUN
ap-1412	158	24	=	=	SYM
ap-1412	158	25	q	q	NOUN
ap-1412	158	26	}	}	PUNCT
ap-1412	158	27	.	.	PUNCT
ap-1412	159	1	then	then	ADV
ap-1412	159	2	(	(	PUNCT
ap-1412	159	3	[	[	X
ap-1412	159	4	0	0	NUM
ap-1412	159	5	,	,	PUNCT
ap-1412	159	6	q]v(h);⊕q	q]v(h);⊕q	PROPN
ap-1412	159	7	,	,	PUNCT
ap-1412	159	8	0	0	NUM
ap-1412	159	9	,	,	PUNCT
ap-1412	159	10	q	q	X
ap-1412	159	11	)	)	PUNCT
ap-1412	159	12	is	be	AUX
ap-1412	159	13	an	an	DET
ap-1412	159	14	effect	effect	NOUN
ap-1412	159	15	algebra	algebra	NOUN
ap-1412	159	16	with	with	ADP
ap-1412	159	17	unit	unit	NOUN
ap-1412	159	18	q	q	PROPN
ap-1412	159	19	and	and	CCONJ
ap-1412	159	20	a	a	DET
ap-1412	159	21	partial	partial	ADJ
ap-1412	159	22	binary	binary	ADJ
ap-1412	159	23	operation	operation	NOUN
ap-1412	159	24	⊕q	⊕q	PROPN
ap-1412	159	25	defined	define	VERB
ap-1412	159	26	in	in	ADP
ap-1412	159	27	theorem	theorem	ADJ
ap-1412	159	28	3.6	3.6	NUM
ap-1412	159	29	.	.	PUNCT
ap-1412	160	1	remark	remark	VERB
ap-1412	160	2	3.7	3.7	NUM
ap-1412	160	3	(	(	PUNCT
ap-1412	160	4	i	i	NOUN
ap-1412	160	5	)	)	PUNCT
ap-1412	160	6	if	if	SCONJ
ap-1412	160	7	q	q	X
ap-1412	160	8	∈	∈	PROPN
ap-1412	160	9	sp(h	sp(h	X
ap-1412	160	10	)	)	PUNCT
ap-1412	160	11	is	be	AUX
ap-1412	160	12	a	a	DET
ap-1412	160	13	bounded	bounded	ADJ
ap-1412	160	14	operator	operator	NOUN
ap-1412	160	15	then	then	ADV
ap-1412	160	16	[	[	X
ap-1412	160	17	0	0	NUM
ap-1412	160	18	,	,	PUNCT
ap-1412	160	19	q]sp(h	q]sp(h	ADV
ap-1412	160	20	)	)	PUNCT
ap-1412	160	21	=	=	PUNCT
ap-1412	161	1	[	[	X
ap-1412	161	2	0	0	NUM
ap-1412	161	3	,	,	PUNCT
ap-1412	161	4	q]v(h	q]v(h	NOUN
ap-1412	161	5	)	)	PUNCT
ap-1412	162	1	and	and	CCONJ
ap-1412	162	2	it	it	PRON
ap-1412	162	3	is	be	AUX
ap-1412	162	4	an	an	DET
ap-1412	162	5	effect	effect	NOUN
ap-1412	162	6	algebra	algebra	NOUN
ap-1412	162	7	of	of	ADP
ap-1412	162	8	all	all	DET
ap-1412	162	9	bounded	bounded	ADJ
ap-1412	162	10	self	self	NOUN
ap-1412	162	11	-	-	PUNCT
ap-1412	162	12	adjoint	adjoint	NOUN
ap-1412	162	13	positive	positive	ADJ
ap-1412	162	14	operators	operator	NOUN
ap-1412	162	15	between	between	ADP
ap-1412	162	16	0	0	NUM
ap-1412	162	17	and	and	CCONJ
ap-1412	162	18	q	q	X
ap-1412	162	19	(	(	PUNCT
ap-1412	162	20	with	with	ADP
ap-1412	162	21	domain	domain	NOUN
ap-1412	162	22	h	h	NOUN
ap-1412	162	23	)	)	PUNCT
ap-1412	162	24	.	.	PUNCT
ap-1412	163	1	moreover	moreover	ADV
ap-1412	163	2	,	,	PUNCT
ap-1412	163	3	⊕q	⊕q	PROPN
ap-1412	163	4	coincides	coincide	VERB
ap-1412	163	5	with	with	ADP
ap-1412	163	6	the	the	DET
ap-1412	163	7	usual	usual	ADJ
ap-1412	163	8	sum	sum	NOUN
ap-1412	163	9	of	of	ADP
ap-1412	163	10	operators	operator	NOUN
ap-1412	163	11	if	if	SCONJ
ap-1412	163	12	a	a	DET
ap-1412	163	13	⊕q	⊕q	NOUN
ap-1412	163	14	b	b	PROPN
ap-1412	163	15	exists	exist	VERB
ap-1412	163	16	in	in	ADP
ap-1412	163	17	[	[	X
ap-1412	163	18	0	0	NUM
ap-1412	163	19	,	,	PUNCT
ap-1412	163	20	q]sp(h	q]sp(h	ADV
ap-1412	163	21	)	)	PUNCT
ap-1412	163	22	.	.	PUNCT
ap-1412	164	1	(	(	PUNCT
ap-1412	164	2	ii	ii	X
ap-1412	164	3	)	)	PUNCT
ap-1412	164	4	it	it	PRON
ap-1412	164	5	follows	follow	VERB
ap-1412	164	6	from	from	ADP
ap-1412	164	7	(	(	PUNCT
ap-1412	164	8	i	i	NOUN
ap-1412	164	9	)	)	PUNCT
ap-1412	164	10	that	that	SCONJ
ap-1412	164	11	if	if	SCONJ
ap-1412	164	12	q	q	NOUN
ap-1412	164	13	=	=	VERB
ap-1412	164	14	i	i	PRON
ap-1412	164	15	(	(	PUNCT
ap-1412	164	16	the	the	DET
ap-1412	164	17	identity	identity	NOUN
ap-1412	164	18	operator	operator	NOUN
ap-1412	164	19	with	with	ADP
ap-1412	164	20	domain	domain	NOUN
ap-1412	164	21	h	h	NOUN
ap-1412	164	22	)	)	PUNCT
ap-1412	164	23	then	then	ADV
ap-1412	164	24	[	[	X
ap-1412	164	25	0	0	NUM
ap-1412	164	26	,	,	PUNCT
ap-1412	164	27	q]sp(h	q]sp(h	ADV
ap-1412	164	28	)	)	PUNCT
ap-1412	164	29	=	=	PUNCT
ap-1412	165	1	[	[	X
ap-1412	165	2	0	0	NUM
ap-1412	165	3	,	,	PUNCT
ap-1412	165	4	q]v(h	q]v(h	NOUN
ap-1412	165	5	)	)	PUNCT
ap-1412	165	6	=	=	SYM
ap-1412	165	7	e(h	e(h	X
ap-1412	165	8	)	)	PUNCT
ap-1412	165	9	is	be	AUX
ap-1412	165	10	the	the	DET
ap-1412	165	11	hilbert	hilbert	PROPN
ap-1412	165	12	space	space	NOUN
ap-1412	165	13	effect	effect	NOUN
ap-1412	165	14	algebra	algebra	NOUN
ap-1412	165	15	of	of	ADP
ap-1412	165	16	all	all	DET
ap-1412	165	17	self	self	NOUN
ap-1412	165	18	-	-	PUNCT
ap-1412	165	19	adjoint	adjoint	NOUN
ap-1412	165	20	operators	operator	NOUN
ap-1412	165	21	between	between	ADP
ap-1412	165	22	0	0	NUM
ap-1412	165	23	and	and	CCONJ
ap-1412	165	24	the	the	DET
ap-1412	165	25	identity	identity	NOUN
ap-1412	165	26	operator	operator	NOUN
ap-1412	165	27	i	i	PRON
ap-1412	165	28	(	(	PUNCT
ap-1412	165	29	see	see	VERB
ap-1412	165	30	[	[	X
ap-1412	165	31	5	5	NUM
ap-1412	165	32	]	]	NUM
ap-1412	165	33	)	)	PUNCT
ap-1412	165	34	.	.	PUNCT
ap-1412	166	1	(	(	PUNCT
ap-1412	166	2	iii	iii	X
ap-1412	166	3	)	)	PUNCT
ap-1412	166	4	if	if	SCONJ
ap-1412	166	5	q	q	X
ap-1412	166	6	∈	∈	PROPN
ap-1412	166	7	sp(h	sp(h	X
ap-1412	166	8	)	)	PUNCT
ap-1412	166	9	is	be	AUX
ap-1412	166	10	an	an	DET
ap-1412	166	11	unbounded	unbounded	ADJ
ap-1412	166	12	operator	operator	NOUN
ap-1412	166	13	with	with	ADP
ap-1412	166	14	d(q	d(q	PROPN
ap-1412	166	15	)	)	PUNCT
ap-1412	166	16	=	=	SYM
ap-1412	167	1	h	h	NOUN
ap-1412	167	2	then	then	ADV
ap-1412	167	3	every	every	DET
ap-1412	167	4	unbounded	unbounded	ADJ
ap-1412	167	5	operator	operator	NOUN
ap-1412	167	6	a	a	DET
ap-1412	167	7	∈	∈	NOUN
ap-1412	168	1	[	[	X
ap-1412	168	2	0	0	NUM
ap-1412	168	3	,	,	PUNCT
ap-1412	168	4	q]sp(h	q]sp(h	X
ap-1412	168	5	)	)	PUNCT
ap-1412	168	6	has	have	VERB
ap-1412	168	7	d(a	d(a	PROPN
ap-1412	168	8	)	)	PUNCT
ap-1412	168	9	=	=	SYM
ap-1412	169	1	d(q	d(q	PROPN
ap-1412	169	2	)	)	PUNCT
ap-1412	169	3	,	,	PUNCT
ap-1412	169	4	since	since	SCONJ
ap-1412	169	5	then	then	ADV
ap-1412	169	6	there	there	PRON
ap-1412	169	7	exists	exist	VERB
ap-1412	169	8	a	a	DET
ap-1412	169	9	bounded	bounded	ADJ
ap-1412	169	10	operator	operator	NOUN
ap-1412	169	11	c	c	PROPN
ap-1412	169	12	∈	∈	PROPN
ap-1412	169	13	sp(h	sp(h	X
ap-1412	169	14	)	)	PUNCT
ap-1412	169	15	(	(	PUNCT
ap-1412	169	16	hence	hence	ADV
ap-1412	169	17	d(c	d(c	PROPN
ap-1412	169	18	)	)	PUNCT
ap-1412	170	1	=	=	SYM
ap-1412	170	2	h	h	X
ap-1412	170	3	)	)	PUNCT
ap-1412	170	4	such	such	ADJ
ap-1412	170	5	that	that	SCONJ
ap-1412	170	6	a	a	DET
ap-1412	170	7	+	+	NOUN
ap-1412	170	8	c	c	NOUN
ap-1412	170	9	=	=	SYM
ap-1412	170	10	q.	q.	PROPN
ap-1412	170	11	(	(	PUNCT
ap-1412	170	12	iv	iv	X
ap-1412	170	13	)	)	PUNCT
ap-1412	170	14	if	if	SCONJ
ap-1412	170	15	q	q	PROPN
ap-1412	170	16	∈	∈	PROPN
ap-1412	170	17	sp(h	sp(h	X
ap-1412	170	18	)	)	PUNCT
ap-1412	170	19	is	be	AUX
ap-1412	170	20	an	an	DET
ap-1412	170	21	unbounded	unbounded	ADJ
ap-1412	170	22	self	self	NOUN
ap-1412	170	23	-	-	PUNCT
ap-1412	170	24	adjoint	adjoint	NOUN
ap-1412	170	25	operator	operator	NOUN
ap-1412	170	26	then	then	ADV
ap-1412	170	27	(	(	PUNCT
ap-1412	170	28	2q)∗	2q)∗	NUM
ap-1412	170	29	=	=	SYM
ap-1412	170	30	2q∗	2q∗	NUM
ap-1412	170	31	=	=	SYM
ap-1412	170	32	2q	2q	NUM
ap-1412	170	33	∈	∈	NOUN
ap-1412	170	34	sp(h	sp(h	NOUN
ap-1412	170	35	)	)	PUNCT
ap-1412	170	36	.	.	PUNCT
ap-1412	171	1	in	in	ADP
ap-1412	171	2	this	this	DET
ap-1412	171	3	case	case	NOUN
ap-1412	171	4	for	for	ADP
ap-1412	171	5	any	any	DET
ap-1412	171	6	operators	operator	NOUN
ap-1412	171	7	a	a	DET
ap-1412	171	8	,	,	PUNCT
ap-1412	171	9	b	b	X
ap-1412	171	10	∈	∈	PROPN
ap-1412	172	1	[	[	X
ap-1412	172	2	0	0	NUM
ap-1412	172	3	,	,	PUNCT
ap-1412	172	4	q]sp(h	q]sp(h	ADV
ap-1412	172	5	)	)	PUNCT
ap-1412	172	6	one	one	PRON
ap-1412	172	7	has	have	VERB
ap-1412	172	8	a	a	DET
ap-1412	172	9	+	+	NOUN
ap-1412	172	10	b	b	NOUN
ap-1412	172	11	∈	∈	NOUN
ap-1412	172	12	sp(h	sp(h	X
ap-1412	172	13	)	)	PUNCT
ap-1412	172	14	(	(	PUNCT
ap-1412	172	15	the	the	DET
ap-1412	172	16	usual	usual	ADJ
ap-1412	172	17	sum	sum	NOUN
ap-1412	172	18	of	of	ADP
ap-1412	172	19	operators	operator	NOUN
ap-1412	172	20	)	)	PUNCT
ap-1412	172	21	,	,	PUNCT
ap-1412	172	22	even	even	ADV
ap-1412	172	23	if	if	SCONJ
ap-1412	172	24	a	a	DET
ap-1412	172	25	,	,	PUNCT
ap-1412	172	26	b	b	NOUN
ap-1412	172	27	are	be	AUX
ap-1412	172	28	unbounded	unbounded	ADJ
ap-1412	172	29	.	.	PUNCT
ap-1412	173	1	the	the	DET
ap-1412	173	2	last	last	NOUN
ap-1412	173	3	follows	follow	VERB
ap-1412	173	4	from	from	ADP
ap-1412	173	5	the	the	DET
ap-1412	173	6	fact	fact	NOUN
ap-1412	173	7	that	that	SCONJ
ap-1412	173	8	there	there	PRON
ap-1412	173	9	are	be	VERB
ap-1412	173	10	bounded	bounded	ADJ
ap-1412	173	11	operators	operator	NOUN
ap-1412	173	12	ca	can	AUX
ap-1412	173	13	,	,	PUNCT
ap-1412	173	14	cb	cb	PROPN
ap-1412	173	15	∈	∈	PROPN
ap-1412	173	16	sp(h	sp(h	NOUN
ap-1412	173	17	)	)	PUNCT
ap-1412	173	18	such	such	ADJ
ap-1412	173	19	that	that	DET
ap-1412	173	20	q	q	NOUN
ap-1412	173	21	=	=	PUNCT
ap-1412	173	22	a	a	DET
ap-1412	173	23	⊕ca	⊕ca	NOUN
ap-1412	173	24	=	=	SYM
ap-1412	173	25	b	b	PROPN
ap-1412	173	26	⊕	⊕	PROPN
ap-1412	173	27	cb	cb	PROPN
ap-1412	173	28	.	.	PUNCT
ap-1412	174	1	thus	thus	ADV
ap-1412	174	2	(	(	PUNCT
ap-1412	174	3	a	a	DET
ap-1412	174	4	⊕	⊕	PROPN
ap-1412	174	5	ca	ca	NOUN
ap-1412	174	6	)	)	PUNCT
ap-1412	175	1	+	+	CCONJ
ap-1412	175	2	(	(	PUNCT
ap-1412	175	3	b	b	PROPN
ap-1412	175	4	⊕	⊕	PROPN
ap-1412	175	5	cb	cb	PROPN
ap-1412	175	6	)	)	PUNCT
ap-1412	175	7	=	=	SYM
ap-1412	175	8	2q	2q	NOUN
ap-1412	175	9	,	,	PUNCT
ap-1412	175	10	hence	hence	ADV
ap-1412	175	11	(	(	PUNCT
ap-1412	175	12	a	a	DET
ap-1412	175	13	+	+	NOUN
ap-1412	175	14	b	b	NOUN
ap-1412	175	15	)	)	PUNCT
ap-1412	176	1	+	+	CCONJ
ap-1412	176	2	(	(	PUNCT
ap-1412	176	3	ca	ca	AUX
ap-1412	176	4	+	+	CCONJ
ap-1412	176	5	cb	cb	PROPN
ap-1412	176	6	)	)	PUNCT
ap-1412	176	7	=	=	SYM
ap-1412	176	8	2q	2q	X
ap-1412	176	9	.	.	PUNCT
ap-1412	177	1	here	here	ADV
ap-1412	177	2	ca+cb	ca+cb	VERB
ap-1412	177	3	∈	∈	PROPN
ap-1412	177	4	sp(h	sp(h	NOUN
ap-1412	177	5	)	)	PUNCT
ap-1412	177	6	and	and	CCONJ
ap-1412	177	7	because	because	SCONJ
ap-1412	177	8	sp(h	sp(h	NOUN
ap-1412	177	9	)	)	PUNCT
ap-1412	177	10	is	be	AUX
ap-1412	177	11	a	a	DET
ap-1412	177	12	generalized	generalized	ADJ
ap-1412	177	13	effect	effect	NOUN
ap-1412	177	14	algebra	algebra	NOUN
ap-1412	177	15	and	and	CCONJ
ap-1412	177	16	also	also	ADV
ap-1412	177	17	2q	2q	NUM
ap-1412	177	18	∈	∈	NOUN
ap-1412	177	19	sp(h	sp(h	X
ap-1412	177	20	)	)	PUNCT
ap-1412	177	21	we	we	PRON
ap-1412	177	22	obtain	obtain	VERB
ap-1412	177	23	that	that	SCONJ
ap-1412	177	24	a	a	DET
ap-1412	177	25	+	+	NOUN
ap-1412	177	26	b	b	NOUN
ap-1412	177	27	∈	∈	NOUN
ap-1412	177	28	sp(h	sp(h	NOUN
ap-1412	177	29	)	)	PUNCT
ap-1412	177	30	.	.	PUNCT
ap-1412	178	1	(	(	PUNCT
ap-1412	178	2	v	v	X
ap-1412	178	3	)	)	PUNCT
ap-1412	178	4	it	it	PRON
ap-1412	178	5	is	be	AUX
ap-1412	178	6	worth	worth	ADJ
ap-1412	178	7	noting	note	VERB
ap-1412	178	8	that	that	SCONJ
ap-1412	178	9	effect	effect	NOUN
ap-1412	178	10	algebras	algebra	NOUN
ap-1412	178	11	are	be	AUX
ap-1412	178	12	very	very	ADV
ap-1412	178	13	natural	natural	ADJ
ap-1412	178	14	structures	structure	NOUN
ap-1412	178	15	as	as	ADP
ap-1412	178	16	carriers	carrier	NOUN
ap-1412	178	17	of	of	ADP
ap-1412	178	18	states	state	NOUN
ap-1412	178	19	(	(	PUNCT
ap-1412	178	20	or	or	CCONJ
ap-1412	178	21	probability	probability	NOUN
ap-1412	178	22	measures	measure	NOUN
ap-1412	178	23	)	)	PUNCT
ap-1412	178	24	when	when	SCONJ
ap-1412	178	25	we	we	PRON
ap-1412	178	26	handle	handle	VERB
ap-1412	178	27	also	also	ADV
ap-1412	178	28	noncompatible	noncompatible	ADJ
ap-1412	178	29	pairs	pair	NOUN
ap-1412	178	30	or	or	CCONJ
ap-1412	178	31	unsharp	unsharp	ADJ
ap-1412	178	32	elements	element	NOUN
ap-1412	178	33	.	.	PUNCT
ap-1412	179	1	acknowledgement	acknowledgement	NOUN
ap-1412	179	2	supported	support	VERB
ap-1412	179	3	by	by	ADP
ap-1412	179	4	vega	vega	PROPN
ap-1412	179	5	1/0297/11	1/0297/11	PROPN
ap-1412	179	6	grant	grant	NOUN
ap-1412	179	7	of	of	ADP
ap-1412	179	8	the	the	DET
ap-1412	179	9	ministry	ministry	PROPN
ap-1412	179	10	of	of	ADP
ap-1412	179	11	education	education	NOUN
ap-1412	179	12	of	of	ADP
ap-1412	179	13	the	the	DET
ap-1412	179	14	slovak	slovak	ADJ
ap-1412	179	15	republic	republic	NOUN
ap-1412	179	16	.	.	PUNCT
ap-1412	180	1	references	reference	NOUN
ap-1412	180	2	[	[	X
ap-1412	180	3	1	1	NUM
ap-1412	180	4	]	]	X
ap-1412	180	5	bagarello	bagarello	PROPN
ap-1412	180	6	,	,	PUNCT
ap-1412	180	7	f.	f.	PROPN
ap-1412	180	8	:	:	PUNCT
ap-1412	180	9	algebras	algebra	NOUN
ap-1412	180	10	of	of	ADP
ap-1412	180	11	unbounded	unbounded	ADJ
ap-1412	180	12	operators	operator	NOUN
ap-1412	180	13	and	and	CCONJ
ap-1412	180	14	physical	physical	ADJ
ap-1412	180	15	applications	application	NOUN
ap-1412	180	16	:	:	PUNCT
ap-1412	180	17	a	a	DET
ap-1412	180	18	survey	survey	NOUN
ap-1412	180	19	,	,	PUNCT
ap-1412	180	20	reviews	review	NOUN
ap-1412	180	21	in	in	ADP
ap-1412	180	22	mathematical	mathematical	ADJ
ap-1412	180	23	physics	physics	NOUN
ap-1412	180	24	19	19	NUM
ap-1412	180	25	(	(	PUNCT
ap-1412	180	26	2007	2007	NUM
ap-1412	180	27	)	)	PUNCT
ap-1412	180	28	,	,	PUNCT
ap-1412	180	29	231–271	231–271	NUM
ap-1412	180	30	.	.	PUNCT
ap-1412	181	1	[	[	X
ap-1412	181	2	2	2	NUM
ap-1412	181	3	]	]	SYM
ap-1412	181	4	birkhoff	birkhoff	NOUN
ap-1412	181	5	,	,	PUNCT
ap-1412	181	6	g.	g.	PROPN
ap-1412	181	7	,	,	PUNCT
ap-1412	181	8	von	von	PROPN
ap-1412	181	9	neumann	neumann	PROPN
ap-1412	181	10	,	,	PUNCT
ap-1412	181	11	j.	j.	PROPN
ap-1412	181	12	:	:	PUNCT
ap-1412	181	13	the	the	DET
ap-1412	181	14	logic	logic	NOUN
ap-1412	181	15	of	of	ADP
ap-1412	181	16	quantum	quantum	ADJ
ap-1412	181	17	mechanics	mechanic	NOUN
ap-1412	181	18	,	,	PUNCT
ap-1412	181	19	ann	ann	PROPN
ap-1412	181	20	.	.	PROPN
ap-1412	181	21	math	math	PROPN
ap-1412	181	22	.	.	PUNCT
ap-1412	182	1	37	37	NUM
ap-1412	182	2	(	(	PUNCT
ap-1412	182	3	1936	1936	NUM
ap-1412	182	4	)	)	PUNCT
ap-1412	182	5	,	,	PUNCT
ap-1412	182	6	823–843	823–843	NUM
ap-1412	182	7	.	.	PUNCT
ap-1412	183	1	[	[	X
ap-1412	183	2	3	3	NUM
ap-1412	183	3	]	]	X
ap-1412	183	4	blank	blank	NOUN
ap-1412	183	5	,	,	PUNCT
ap-1412	183	6	j.	j.	PROPN
ap-1412	183	7	,	,	PUNCT
ap-1412	183	8	exner	exner	PROPN
ap-1412	183	9	,	,	PUNCT
ap-1412	183	10	p.	p.	PROPN
ap-1412	183	11	,	,	PUNCT
ap-1412	183	12	havĺıček	havĺıček	PROPN
ap-1412	183	13	,	,	PUNCT
ap-1412	183	14	m.	m.	NOUN
ap-1412	183	15	:	:	PUNCT
ap-1412	183	16	hilbert	hilbert	NOUN
ap-1412	183	17	space	space	NOUN
ap-1412	183	18	operators	operator	NOUN
ap-1412	183	19	in	in	ADP
ap-1412	183	20	quantum	quantum	ADJ
ap-1412	183	21	physics	physics	NOUN
ap-1412	183	22	.	.	PUNCT
ap-1412	184	1	(	(	PUNCT
ap-1412	184	2	second	second	ADJ
ap-1412	184	3	edition	edition	NOUN
ap-1412	184	4	)	)	PUNCT
ap-1412	184	5	,	,	PUNCT
ap-1412	184	6	springer	springer	NOUN
ap-1412	184	7	,	,	PUNCT
ap-1412	184	8	2008	2008	NUM
ap-1412	184	9	.	.	PUNCT
ap-1412	185	1	[	[	X
ap-1412	185	2	4	4	NUM
ap-1412	185	3	]	]	X
ap-1412	185	4	dvurečenskij	dvurečenskij	PROPN
ap-1412	185	5	,	,	PUNCT
ap-1412	185	6	a.	a.	NOUN
ap-1412	185	7	,	,	PUNCT
ap-1412	185	8	pulmannová	pulmannová	ADJ
ap-1412	185	9	,	,	PUNCT
ap-1412	185	10	s.	s.	PROPN
ap-1412	185	11	:	:	PUNCT
ap-1412	185	12	new	new	ADJ
ap-1412	185	13	trends	trend	NOUN
ap-1412	185	14	in	in	ADP
ap-1412	185	15	quantum	quantum	ADJ
ap-1412	185	16	structures	structure	NOUN
ap-1412	185	17	.	.	PUNCT
ap-1412	186	1	dodrecht	dodrecht	NOUN
ap-1412	186	2	:	:	PUNCT
ap-1412	186	3	kluwer	kluwer	NOUN
ap-1412	186	4	,	,	PUNCT
ap-1412	186	5	the	the	DET
ap-1412	186	6	netherlands	netherlands	PROPN
ap-1412	186	7	,	,	PUNCT
ap-1412	186	8	2000	2000	NUM
ap-1412	186	9	.	.	PUNCT
ap-1412	187	1	[	[	X
ap-1412	187	2	5	5	NUM
ap-1412	187	3	]	]	X
ap-1412	187	4	foulis	foulis	PROPN
ap-1412	187	5	,	,	PUNCT
ap-1412	187	6	d.	d.	PROPN
ap-1412	187	7	j.	j.	PROPN
ap-1412	187	8	,	,	PUNCT
ap-1412	187	9	bennett	bennett	PROPN
ap-1412	187	10	,	,	PUNCT
ap-1412	187	11	m.	m.	PROPN
ap-1412	187	12	k.	k.	PROPN
ap-1412	187	13	:	:	PUNCT
ap-1412	187	14	effect	effect	NOUN
ap-1412	187	15	algebras	algebra	NOUN
ap-1412	187	16	and	and	CCONJ
ap-1412	187	17	unsharp	unsharp	ADJ
ap-1412	187	18	quantum	quantum	ADJ
ap-1412	187	19	logics	logic	NOUN
ap-1412	187	20	,	,	PUNCT
ap-1412	187	21	found	find	VERB
ap-1412	187	22	.	.	PUNCT
ap-1412	188	1	phys	phy	NOUN
ap-1412	188	2	.	.	PUNCT
ap-1412	189	1	24	24	NUM
ap-1412	189	2	(	(	PUNCT
ap-1412	189	3	1994	1994	NUM
ap-1412	189	4	)	)	PUNCT
ap-1412	189	5	,	,	PUNCT
ap-1412	189	6	1	1	NUM
ap-1412	189	7	331–1352	331–1352	NUM
ap-1412	189	8	.	.	PUNCT
ap-1412	190	1	[	[	X
ap-1412	190	2	6	6	NUM
ap-1412	190	3	]	]	PUNCT
ap-1412	190	4	hedĺıková	hedĺıková	PROPN
ap-1412	190	5	,	,	PUNCT
ap-1412	190	6	j.	j.	PROPN
ap-1412	190	7	,	,	PUNCT
ap-1412	190	8	pulmannová	pulmannová	ADV
ap-1412	190	9	,	,	PUNCT
ap-1412	190	10	s.	s.	PROPN
ap-1412	190	11	:	:	PUNCT
ap-1412	190	12	generalized	generalized	ADJ
ap-1412	190	13	difference	difference	NOUN
ap-1412	190	14	posets	poset	NOUN
ap-1412	190	15	and	and	CCONJ
ap-1412	190	16	orthoalgebras	orthoalgebra	NOUN
ap-1412	190	17	,	,	PUNCT
ap-1412	190	18	acta	acta	PROPN
ap-1412	190	19	math	math	PROPN
ap-1412	190	20	.	.	PUNCT
ap-1412	191	1	univ	univ	PROPN
ap-1412	191	2	.	.	PROPN
ap-1412	191	3	comenianae	comenianae	PROPN
ap-1412	191	4	lxv	lxv	PROPN
ap-1412	191	5	(	(	PUNCT
ap-1412	191	6	1996	1996	NUM
ap-1412	191	7	)	)	PUNCT
ap-1412	191	8	,	,	PUNCT
ap-1412	191	9	247–279	247–279	NUM
ap-1412	191	10	.	.	PUNCT
ap-1412	192	1	[	[	X
ap-1412	192	2	7	7	X
ap-1412	192	3	]	]	X
ap-1412	192	4	kalmbach	kalmbach	NOUN
ap-1412	192	5	,	,	PUNCT
ap-1412	192	6	g.	g.	PROPN
ap-1412	192	7	,	,	PUNCT
ap-1412	192	8	riečanová	riečanová	PROPN
ap-1412	192	9	,	,	PUNCT
ap-1412	192	10	z.	z.	PROPN
ap-1412	192	11	:	:	PUNCT
ap-1412	192	12	an	an	DET
ap-1412	192	13	axiomatization	axiomatization	NOUN
ap-1412	192	14	for	for	ADP
ap-1412	192	15	abelian	abelian	PROPN
ap-1412	192	16	relative	relative	PROPN
ap-1412	192	17	inverses	inverses	PROPN
ap-1412	192	18	,	,	PUNCT
ap-1412	192	19	demonstratio	demonstratio	PROPN
ap-1412	192	20	math	math	PROPN
ap-1412	192	21	.	.	PUNCT
ap-1412	193	1	27	27	NUM
ap-1412	193	2	(	(	PUNCT
ap-1412	193	3	1994	1994	NUM
ap-1412	193	4	)	)	PUNCT
ap-1412	193	5	,	,	PUNCT
ap-1412	193	6	769–780	769–780	NUM
ap-1412	193	7	.	.	PUNCT
ap-1412	194	1	[	[	X
ap-1412	194	2	8	8	NUM
ap-1412	194	3	]	]	X
ap-1412	194	4	kôpka	kôpka	NOUN
ap-1412	194	5	,	,	PUNCT
ap-1412	194	6	f.	f.	PROPN
ap-1412	194	7	,	,	PUNCT
ap-1412	194	8	chovanec	chovanec	PROPN
ap-1412	194	9	,	,	PUNCT
ap-1412	194	10	f.	f.	PROPN
ap-1412	194	11	:	:	PUNCT
ap-1412	194	12	d	d	X
ap-1412	194	13	-	-	PUNCT
ap-1412	194	14	posets	poset	NOUN
ap-1412	194	15	,	,	PUNCT
ap-1412	194	16	math	math	NOUN
ap-1412	194	17	.	.	PUNCT
ap-1412	195	1	slovaca	slovaca	NOUN
ap-1412	195	2	44	44	NUM
ap-1412	195	3	(	(	PUNCT
ap-1412	195	4	1994	1994	NUM
ap-1412	195	5	)	)	PUNCT
ap-1412	195	6	,	,	PUNCT
ap-1412	195	7	21–34	21–34	NUM
ap-1412	195	8	.	.	PUNCT
ap-1412	196	1	[	[	X
ap-1412	196	2	9	9	NUM
ap-1412	196	3	]	]	SYM
ap-1412	196	4	polakovič	polakovič	NOUN
ap-1412	196	5	,	,	PUNCT
ap-1412	196	6	m.	m.	NOUN
ap-1412	196	7	,	,	PUNCT
ap-1412	196	8	riečanová	riečanová	PROPN
ap-1412	196	9	,	,	PUNCT
ap-1412	196	10	z.	z.	PROPN
ap-1412	196	11	:	:	PUNCT
ap-1412	196	12	generalized	generalized	ADJ
ap-1412	196	13	effect	effect	NOUN
ap-1412	196	14	algebras	algebra	NOUN
ap-1412	196	15	of	of	ADP
ap-1412	196	16	positive	positive	ADJ
ap-1412	196	17	operators	operator	NOUN
ap-1412	196	18	densely	densely	ADV
ap-1412	196	19	defined	define	VERB
ap-1412	196	20	on	on	ADP
ap-1412	196	21	hilbert	hilbert	NOUN
ap-1412	196	22	space	space	NOUN
ap-1412	196	23	,	,	PUNCT
ap-1412	196	24	internat	internat	PROPN
ap-1412	196	25	.	.	PUNCT
ap-1412	197	1	j.	j.	PROPN
ap-1412	197	2	theor	theor	PROPN
ap-1412	197	3	.	.	PUNCT
ap-1412	198	1	phys	phy	NOUN
ap-1412	198	2	50	50	NUM
ap-1412	198	3	(	(	PUNCT
ap-1412	198	4	2011	2011	NUM
ap-1412	198	5	)	)	PUNCT
ap-1412	198	6	,	,	PUNCT
ap-1412	198	7	1	1	NUM
ap-1412	198	8	167–1	167–1	NUM
ap-1412	198	9	174	174	NUM
ap-1412	198	10	.	.	PUNCT
ap-1412	199	1	81	81	NUM
ap-1412	199	2	acta	acta	PROPN
ap-1412	199	3	polytechnica	polytechnica	PROPN
ap-1412	199	4	vol	vol	NOUN
ap-1412	199	5	.	.	PUNCT
ap-1412	200	1	51	51	NUM
ap-1412	200	2	no	no	INTJ
ap-1412	200	3	.	.	PUNCT
ap-1412	200	4	4/2011	4/2011	NUM
ap-1412	201	1	[	[	SYM
ap-1412	201	2	10	10	NUM
ap-1412	201	3	]	]	X
ap-1412	201	4	reed	reed	NOUN
ap-1412	201	5	,	,	PUNCT
ap-1412	201	6	m.	m.	NOUN
ap-1412	201	7	,	,	PUNCT
ap-1412	201	8	simon	simon	PROPN
ap-1412	201	9	,	,	PUNCT
ap-1412	201	10	b.	b.	PROPN
ap-1412	201	11	:	:	PUNCT
ap-1412	201	12	methods	method	NOUN
ap-1412	201	13	of	of	ADP
ap-1412	201	14	modern	modern	ADJ
ap-1412	201	15	mathematical	mathematical	ADJ
ap-1412	201	16	physics	physics	PROPN
ap-1412	201	17	ii	ii	PROPN
ap-1412	201	18	,	,	PUNCT
ap-1412	201	19	fourier	fourier	ADJ
ap-1412	201	20	analysis	analysis	NOUN
ap-1412	201	21	,	,	PUNCT
ap-1412	201	22	selfadjointness	selfadjointness	NOUN
ap-1412	201	23	.	.	PUNCT
ap-1412	202	1	new	new	PROPN
ap-1412	202	2	york	york	PROPN
ap-1412	202	3	,	,	PUNCT
ap-1412	202	4	san	san	PROPN
ap-1412	202	5	francisco	francisco	PROPN
ap-1412	202	6	,	,	PUNCT
ap-1412	202	7	london	london	PROPN
ap-1412	202	8	:	:	PUNCT
ap-1412	202	9	academic	academic	ADJ
ap-1412	202	10	press	press	NOUN
ap-1412	202	11	,	,	PUNCT
ap-1412	202	12	1975	1975	NUM
ap-1412	202	13	.	.	PUNCT
ap-1412	203	1	[	[	X
ap-1412	203	2	11	11	NUM
ap-1412	203	3	]	]	SYM
ap-1412	203	4	riečanová	riečanová	PROPN
ap-1412	203	5	,	,	PUNCT
ap-1412	203	6	z.	z.	PROPN
ap-1412	203	7	:	:	PUNCT
ap-1412	203	8	subalgebras	subalgebras	PROPN
ap-1412	203	9	,	,	PUNCT
ap-1412	203	10	intervals	interval	NOUN
ap-1412	203	11	and	and	CCONJ
ap-1412	203	12	central	central	ADJ
ap-1412	203	13	elements	element	NOUN
ap-1412	203	14	of	of	ADP
ap-1412	203	15	generalized	generalized	ADJ
ap-1412	203	16	effect	effect	NOUN
ap-1412	203	17	algebras	algebra	NOUN
ap-1412	203	18	,	,	PUNCT
ap-1412	203	19	international	international	ADJ
ap-1412	203	20	journal	journal	NOUN
ap-1412	203	21	of	of	ADP
ap-1412	203	22	theoretical	theoretical	ADJ
ap-1412	203	23	physics	physics	NOUN
ap-1412	203	24	38	38	NUM
ap-1412	203	25	(	(	PUNCT
ap-1412	203	26	1999	1999	NUM
ap-1412	203	27	)	)	PUNCT
ap-1412	203	28	,	,	PUNCT
ap-1412	203	29	3	3	NUM
ap-1412	203	30	209–3220	209–3220	NUM
ap-1412	203	31	.	.	PUNCT
ap-1412	204	1	zdenka	zdenka	PROPN
ap-1412	204	2	riečanová	riečanová	PROPN
ap-1412	204	3	e	e	PROPN
ap-1412	204	4	-	-	NOUN
ap-1412	204	5	mail	mail	NOUN
ap-1412	204	6	:	:	PUNCT
ap-1412	204	7	zdenka.riecanova@stuba.sk	zdenka.riecanova@stuba.sk	PROPN
ap-1412	204	8	department	department	PROPN
ap-1412	204	9	of	of	ADP
ap-1412	204	10	mathematics	mathematics	PROPN
ap-1412	204	11	faculty	faculty	NOUN
ap-1412	204	12	of	of	ADP
ap-1412	204	13	electrical	electrical	ADJ
ap-1412	204	14	engineering	engineering	NOUN
ap-1412	204	15	and	and	CCONJ
ap-1412	204	16	information	information	NOUN
ap-1412	204	17	technology	technology	NOUN
ap-1412	204	18	stu	stu	PROPN
ap-1412	204	19	ilkovičova	ilkovičova	VERB
ap-1412	204	20	3	3	NUM
ap-1412	204	21	,	,	PUNCT
ap-1412	204	22	sk-81219	sk-81219	ADJ
ap-1412	204	23	bratislava	bratislava	PROPN
ap-1412	204	24	82	82	NUM