Cubo A Moth.cmo ti col Jo urrrnl 
l'ol.05(/\~<U OCTOBER ~oos 

A haracte ri zat ion of Unbounde d Fredholm Operators 

;l/e.rtrnrl cr• C. Ht1111111 
LM,t¡CNl?S. 31 Cl1 em111 JtMcph A i9u1er 

.lfa1se11/e 13-102, 1:ed1'.r f!O. FrYmcc 
a11d ,\/atJ1rnwfl c.~ /J cpclf'f,me111 .. /(cmsru S tat1 l 'rut•e1 "'Y 

M1rn/rn1t1rn. /(S (}6506-e60J:t. USA 
r·111111 l 11rlrlrv .. ~.~ : rrunm«lmath . ksu .edu 

ln t rod u tion and t he Result 

TJ11 .. pnpM 1s n ro111111un 11011 of j I]. wh Ol't' bou ud cd Fredho hu o¡X'rn lol'- an> st \l dir1d . T l1 t' 
1l 1t 'Qt\' uf b uudt,.f lull"a.t Frr<l hohn· t ,Ypr opon1 tor::. b prescm cd m nwoy U?Xt:., ::.ce P.g. [! J. 
12] T hb pApN t- -.·n11t11 for n bl'ond oudic11 cl' a1 1cl t i1l' nuth or 1 ries 10 g1 \-C s imple o.11d s l1or1 
nt~mm• UC .¡ 

Wr rnll 11 lmtor clrucd d1.:.11.scly llcfiu ml opcmtor A : X - ) · achn9 from a B a11t1 ch 81)111:e 
\ mta 11 Oanach pocr 1· a rh:d/10/111 o¡Jt•mlor, ami wnte A E Fnd(X . l .) 1/ arul 011/y ·1f 

11,(A) = R(A ) 

'"'" 
u( ..t ) = 11(/1 ' ) := 11 < 

11hnr N(A) ~· fu , \u • O.u E /J(A)}. 

11 (..I ) := ditu !\ ·( ,. \ ). 

( 11 ) 

( 1. 2) 

111 ihc htrnuur~ llK" Noe1 h r opernLon; t\l"C sometim cs co.ll ed Ftcdhohn opernt.ors. The 
'.:twlht•r opem tON attOJ>('rfl.l or:- for whkl1 ( l . l ) liolds. 11( ;\ ) < . u{.'\ ")< . but r1 {A) 
1111 1\ hl' 110 1 fttUid 10 "(A "). Tlms Frcrl(X. Y) is n propcr s\lbsct of t hc :'\oeth cr opcrntors. 

Th1 • Ncwd1n optTllll Oft(; are coll('(I 111 honor o f F . :'\oc thcr. who wns 1lu;• firsl l o si ud ,Y n 
• lnf>.~ of 111 11¡.¡ulnr 1otiti;.111I rqmu1011s with op('rntors of 1h1 s dnss iu 1921 jJJ. 



92 A CJ11ll'a cterizl.ltior1 o( U11bounded F'rcdl1olm Operntoffl 

In f11j " si mpl e nnd s hort proof of t he Fredho lm ahernat1 \-e and n chnratl riuuon ol 
Fredholm opero t.ors ar e g:iven for boundt..>d linear opem1ors. Rccall thnt " lincnr boundtd 
opern tor F lS ca ll ed a finitc- rank operntor ií dim R{F ) < , whe re R(F ) ili th e rftng~ of F 

In t he presc nt. papcr ti he resu lts of (4] are generalized to thc en.se of closcd uuboundi!d 
linear opern1ors. Niuncl y, t he following rt.>s ult is pro\•ed: 

Theo r em 1.1 lf A is r1 Frcdholm opcmtor, then 

A = 8 - F, (131 

U!hr:rt: Bu a /mear c/o.rnrl 071emlor, D(B ) = D( A ), R ( B ) = )~. N(B) = {O) , 1md Fu cr 
fimlc-ronk opem tor. Comrnr.~ely, if (1. 3) lwlds. whcnr 13 ·X - )' .... a l111eor closc.d dc.n,,rlr 
defi11ed oparo t-o,.. R {/3) = i' , N(B) = {O) . arid F r.s u fimtc-ronJ.: opttm tor, tl1r; 11 A u rÜ,JcJ 
/J(.-1) = 0(8). and (1. 1) rmd (l. !!} hofd. so A 1s a Fredholm opcmtor. 

lu !>Ce t1011 2 11 proof of T heo rti m 1 is givc n. 111 th c htcraturc 1lu.' co.:.c of unboundtd 
Frcdholm opcrators is usun lly no! d iscusscd dírcc tly. In jSj tmd m [6j. pp.5i· GI. s mgulllfllU"' 
of 1hc pn.r1u nct er-dcpc11dc11!, Frcd holm opcrntors nrc s tud icd. 1111(! iu !7J nppll cn11011?i of tlwo 
Frcdho hn OJ>Crn to rs in brnn chiu g tlhco ry nrc prcsc nt cd Thcorcm 1 1 is UbC Ítll. for cx1u11plr 
111 lhc 1hoory of c l\ipt,ic boundnry vnlu c problcms, but "-"t' do no t go in10 fur1h1•r dl•ltlll (~ 

c.g .. (11. (21. l;j). 

2 Proof 

1 ;\ssi1111c 1hnt A : X - )" i~ lin cnr , close<l, d cnsely defi ned o¡>cra 1or, und ( 1 1) nnd (12) 
hold l..c t lL~ provc 1h111, t1hc11 ( ! .:.!) holds, D ( /J ) = D ( A ). R( B ) = l '.N( B} = fDt . 8111rl~ 
11ml F 1:, fim1c- 11111k opt·1·ntor . 

Lc1 ~;,}i'5'.,'fr1 be n bnsis of N(i\) nnd fl •J }i -5'.J°f" be a bas11> of N(JI º ) h lb k111rrm 
1h8' 

/1 (J\)J. = N (A ' ). (2 1) 

whcre R( ..t)J. 1s !lw set of liticur fun c1io11nl1t h'11 111 1 ~ · sud1 1lia1 (i.-•1, J\11 ) = O V11 1)(11). 
whcrc (-.·,. /) is tl1 c vn lu c of 11 li11 cn r íu11 c110 11nl v, E l .. on lht' e l<'1m·111 f }' Clr1Ul~·. 
t•1 E ,\ '( A") . 1 :S: ; $ 11 . 

Dc-fmt' 

/Ju := Au + L ( /1 J.u)11, ·=( A + f 1u. ,,, E } ". 

'' 
1221 



A .C. H.nmm 93 

"'hnro F In n flnllc-ra.nk opcrnlor, {vJ }i sJ.S n is n aot. of cloments of Y , biorthogo nul to the 

{
O,j ¡é m . 

le ~ ! "1J h SJ 5 111 (~,. u.,.) .. 6Jm := l ,j = m , and {liJ }t SJ~n IS th e: set of elemcnta of 
x·, Ulorthogo11nl 10 t ho set (1PJ)1 ~J.Sn • (l1 J,¡p ,., ) = ÓJ•n · Existence of .!!ets bio rt hogonal 
to flnlt o.ly mnr.\)· hnCMl.y lndo pcmdc.nt ole.menta of n Bnnnch spnce follows from t.he Jfo.lm -
B11 nool1 lhoorom. An ar bilrnry olomcmt 'll E X can be un iquely represc.nt.ed a.s 1J = u 1 + 
I:;'. 1 l°J 'PJ• CJ m t'ODSI, rutd (Ji¡, U¡)= Q, 1 :5 j $ 11. 

Lct ue chedt thft t N(B) = {O} and /l(B) = Y . Ass um e B u = O, t.hnt is Au + 
'f..J'. 1(111 , u}vJ • O. Apply l/'m t.o tlhia oquntiou, uso (rb,,, , A-u)= O, and get 

n " 

O• L(~m,v¡)(h¡, u) = L; 6,,,;(h¡, u) = (h,,,, u), 1 $ m $ n. 
,.. J• l 

1'lwruforc .<l u - o. So 11 e (A) , nnd 'U = Lj'. 1 CJ'P11 Cj = const. Apply hm l;o t hi a 
N¡un tlo n o, nd use (h,..,ip¡) :::z 6,,v to got c.,. =O, 1 ~ m :S n . Thus u= O. We hnve preved 

''"' N(B) • (O}. 

'lb l)rOYtl R( B ) = Y , t.f\ke nn nrbitmry ol ement f E Y and v.'Tite f = Ít + /2, wh ore 
/1 • Au1 b long to R(A), ru id h. ,,. í:j'. 1 a111¡, ªJ = co nst. No te tbat. 

Y = R(A) + L,,, (2.3) 

whtr t ho 1um \$ dirtct , Ln Is spnn ncd by the e lume ntil {v1 } 1 s_1 ~n· and a1 = (1/1;, !). lndecd, 

(-....., ,/ ) = (~1n11 Au.¡) + La¡(,P,., , uJ) = ª"" l :S m :S n . 
J• • 

Clvou nn nrbiirl\l)' /e l', j = A11 1 + EJ1• 1{.,P1,/)111, define u= u1 + L.;. 1(..p1, f) tp;, wher e 
(hJ, 111) • O, 1 $ J $ n Thcn Bu = f . lndcod, uaing (2.2) ouo hns: 

IJ [• • + t. (4';. /l ,] = Au, + :t(/1¡ , u,)v; + :t(/•¡ , :t (~m.f) m)v, = f. (2.4) 
J•\ J• I 1•1 m• I 

Horu ih~ ~t.uc:m.s (h1 ,¡p n, ) = Ój.,. nnd (111, u1) = O are uscd. We: bnve provcd t he 
rt!lnt lon ll(B)= Y 

'l Lo ~ Utl no-o MlllU.tnc tlu:i.t. A • B - P , wh c rc 8 : X __, Y i!i e. linear clost.'d don aoly 
dt!íl n 1 Op~raíoc , D{A) .. D(B), N(B) = {O) , R(B ) =V, o.nd F is B fini1.e-rru1k opc rn tor. 
Wc w\11.h to Prtft't" 1.h.M ( l l) an d ( l .2} hold and A is closcd . 

Lol 1111 pr'O\--e: (1 1). A.ssumo thnt Au,. := fn - f and J)rQ\re that. /E R{ A). 



94 A Charac rerization of Unbounded Ftedliolm Operntors 

One has Bun - Fu" --. f. Since N(B) = {O}, R(B ) = Y, and 8 is closed, a-1 is 
bounded by Banach's theorem. Thus 

(2.5) 

Since F is a fin ite--rank operator, s - 1 F is compact. Therefore, if supn ll tln ll:S e, whcre 
e is a constant , then a subsequence, dcnoted Un again , can be found , such that s - 1 Fun 
conve rges in t he nonn of X. Consequently, (2.5) implies Un -- u, u -B- 1Fu = a- 11, so 
u E D (B ) and Bu- Fu~ f. 

To finish the proof, !et us estab lish the estimate supn Jlunll $c. Assuming llu..~11-- oo 
and denoting nk by n and a- 1F by T , define tln := fu:ir· Uvnll =l. Then Vn -Tun __,o 
as n -- oo. One may as.'3ume that Vn is chosen in a direct complement of N(f - T) in X. 

Arguiug as above , one se lects a convergent in X subsequence, de noted again by Vn, Vn--+ v, 
and gets v - Tu= O. Since v belougs to t he direct complement of N(I - T ), it follows that 
v =O. On th e other hand, si nce llvll = limn ..... 00 llvnll = 1, one gcts a contrndiction , whicl1 
proves the desired estimate sup" llunll ~c. Property ( 1.1 ) is proved . 

Let us prove that A is closed. If Aun-> f and u,...-+ u , then Bun - Fu" -o f, and the 
above argument shows that Bu - Fu= f so Au =f. Thus A is closed. 

Finall y, !et us prove (1.2). 

Let Au =O, i.e. Bu - Fu= O. App lying the bounded linear injective eperator a- 1, 
ene ge t s an equivalent equation 

u - Tu=O , T:=B - 1F, T:X--X, (2 .6) 

with a finite-rank o perator T. It is an elementary fact (see {4 ]) that dim N(J - T) := n < oo 
if T is a fini te-rank operator. Since N(A) = N(I -T), ene has dim N( A ) = n < oo. 

Now Jet Aºv =O. Then 
B ºv-F•v =O. (2.7) 

Since (B º ) - 1 = (B- 1 )º is a bounded and injective linear operater, the e lements v are in 
one-to-o ne corresp ond ence with the elements w := B ºv , aud (2.7) is equiva.lent to 

w - T ' w =O, T' ~ F ' (B ' )-' , (2.8) 

so that T' is the a.djoint to operator T := s -1 F . 
Since T is a finite-rank operator, it is an elementary fact (see [4)) that dim N(l - T ' ) = 

dim(l -T) = n < oo. Since N(A') = N(I -T'), property (1.2) is preved . 
Theerem 1. 1 is preved . o 

An immediate conseque nce of Theorem 1.1 is the Fredholm alternative (see Theorcm 

1.1 in [4]) for unbounded operators A E Fred(X, Y ). 



A.G. Ramm 

References 

[111] KANTOROVJCH, L. ANill AKIL0V, G., F'unctional analysis in normed spaces, Macmillan, 
New York, 1964. 

[2] KNrO, T., Perturbation theory for linear operut<Jrs, Spr.inger Verlag, New York , 1984. 

[3] NOETHER , F., Über eine Kla.sse singuUirer /ntegralgleichungen, Math. Ann. 82 (l921), 

42-63. 

[4.) RAMM, A.G., A simple proof of the Fredholm altemative and a characterization of the 
Fredholm opcmtors, A•mer. Mat'h. M0nthly 108, N9, (20©1), 855-860. 

[5] RAMM , A.G., Singularities of the inverses o/ F're4holm operators, Proc. of Rey. $oc. 

Edinburgh 102A (1986~, 117- 121. 

(6] RAMM, A.G., Scattering by obstacles, Reidel, Dordr-echt, 1986. 

[7) VA!NBERG , M. AND TRENOGIN, V., Thecry of branching of solutions of non linear 

equations, Noordhoff, 1•974. 

95