T H E S P E C T R A L P R O P E R T I E S O F A C E R T A I N

C L A S S O F S E L F - A D J O I N T O P E R A T O R F U N C T I O N S

A . I . V i r o z u b a n d V . I . M a t s a e v

Suppose that L(X) is a holomorphic opera tor function in a domain G and that A is an isolated par t of the spec t rum of LO.). A rramber of works have recently appeared devoted to consider ing the following prob- lem: under what conditions does there exist a bounded, l inear opera tor Z such that the spec t rum of Z coin- cides with A, and the opera tor function L(X)(Z- hi) -1 is holomorphic and invertible on A?

Almost all substantial results in this direct ion re fe r to the case of a Hilbert space and a self-adjoint opera tor function L(X), i .e. , [L(X)]* = L(~).

In 1965 there appeared the work of M. G. Krein and G. Langer [1] devoted t o a sys temat ic study of self-adjoint quadratic pencils (i.e., opera tor polynomials of second order ) . In par t icu lar , the f i r s t impor- tant results pertaining to the problem formulated above were obtained in this paper.

Par t ia l general izat ions of these resul ts to the case of polynomials of a rb i t r a ry o rde r have been ob- tained in the recent papers [2] and [3]. In both these papers it is assumed that L(X) is a polynomial with self-adjoint coefficients and that A is contained in some segment [a, b] of the real axis, whereby L(a) << 0, L(b) >> 0.* Moreover , in [2] it is additionally assumed that L'(x) >> 0 for x E [a, b], and in the paper [3] it is assumed that fo r any v e c t o r f ~ 0 all roots of the polynomial ( L ( M f , f ) are real and only one of these lies on [a, b]. In the papers [2] and [3] it was established that there exists an opera tor Z having the r e - quired proper t ies and that, in addition, it is s imi la r to a self-adjoint opera tor .

Some of the conditions indicated above (namely, the condition of uni form positivity of L'(x) and the condition that all the roots o f ( L ( ~ ) f , f ) be real) s eem to us to be excess ively str ingent . In this paper it will be shown that the aforementioned additional assumptions of the papers [2] and [3] can be replaced by the following condition: there exists a complex neighborhood U of the segment [a, b] such that ( L ( h ) f , ] ) ( f ~ 0) has exactly one root in U. It is easy to see that this condition is more general than the conditions of [2] and [3]. Moreover , the theorem we prove solves the problem not only for the case of a polynomial but for the case of an a rb i t r a ry self-adjoint opera to r function which is holomorphic in U.

The authors express their deepest thanks to A. S. Markus for many valuable d iscuss ions .

§ 1 . F o r m u l a t i o n o f t h e T h e o r e m a n d S o m e R e m a r k s

Let ~ be a Hilbert space, and let • be the set of all bounded, l inear opera tors acting in 9 The spec- t rum of the opera tor A E~ we denote by a(A). We shall wri te A >> 0, if the opera to r A is uniformly posi - tive, i .e. , there exists a rrumber p > 0, such that ( A f , f ) - P ( f , f ) for a l l f E~.

Let B(D be an opera tor function which is holomorphic in some domain G and which takes values in • . The spec t rum of B(D is the set of tt E G such that the opera to r B(~) is not invertible. A point ~ E G is called an eigenvalue of the opera tor function BO,), if there exists a vec tor h ¢ 0 (an eigenvector) such that B(u)h = 0 .

*We wri te A ~> 0 (A << 0), if there exists a number T > 0, such that A ~ TI (A <-- - ~I).

Institute for Chemical Phys ics , Academy of Sciences, USSR. Translated f rom Funktsional 'nyi Analiz i Ego Pr i lozheniya, Vol. 8, No. 1, pp. 1-10, January -March , 1974. Original ar t ic le submitted January 11, 1973.

© 1974 Consultants Bureau, a division of Plenum Publishing Corporation, 227 g~est 17th Street, New York, N. Y. 10011. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the publisher. 3 copy of this article is available from the publisher for $15.00.

If the domain G is symmetric with respect to the real axis and [B(D]* = B(~)(A 6 G), then the opera- tor function B(A) is called self-adjoint in G.

THEOREM. Let [a, b] be a segment of the real axis, let U be a simply connected neighborhood of this segment which is symmetric with respect to the real axis, and let L(A) be an operator function which is holomorphic and self-adjoint in U. If L(a) << 0, L(b) >> 0 and for anyf * 0 the function(LO0f, f) has exaetlyone rootinU which is also simple, then L(M admits the representation

L(~) = L + ( ~ ) ( Z - - ~ I ) ( ~ U ) , (1)

where L+(D is holomorphic and invertible i n U , while Z Eg~ and a(Z) c [a, b]. Moreover , the opera tor Z is s imi la r to self-adjoint opera tor .

COROLLARY 1. L e t ~ be separable and suppose that for some c E [a, b] the opera to r L(c) is com- pletely continuous. Then the eigenvectors of the opera to r function L ( ~ , corresponding to eigenvalues in the segment [a, b] fo rm an unconditional basis in @.

Indeed, f rom the complete continuity of L(c) and Eq. (1) it follows that the opera tor Z - cI is com- pletely continuous, and since this opera to r is s imi la r to a self-adjoint opera to r its e igenvectors form an unconditional basis in @. It remains to r emark that by vir tue of EO. (1) the eigenvalues of the opera tor function L(k) lying in [a, b] and the corresponding e igenvectors coincide with the eigenvalues and eigen- vectors of the ope ra to r Z.

COROLLARY 2. Let V be a neighborhood of the segment [a, b] which is symmet r i c with r e spec t to the rea l axis , let B(?,) be an ope ra to r function which is holomorphic and self-adjoint in V and such that B(x)-> 0, B'(x) -> 0(0 -< x -< b) , and let L(~) = ~ I - - A ÷ ~ B ( ~ ) ( A ~ , A ~ 0 , [ [Al l~b) . Then t h e r e e x - ists a neighborhood U of the segment [0, b] such that

L (~) = L+ (~) (Z -- ~I) (~ ~ U), (2)

where L+(D is holomorphic and invertible in U, the ope ra to r Z is s imi l a r to a nonnegative opera tor , and (z) c [0, U A Ul.

Indeed, it is possible to choose a number a < 0(a E V) such that

[ + z B ( x ) ~ O , I + 2 x B ( x ) +xtB "(x)~>O ( a ~ x ~ O ) ,

and the re fore

L (a) , ~ O, L (b) ~.O, L" (x) ~> O (a ~ x ~ b).

Thus, for a n y f ~ 0 the function ( L ( D f , ] ) has exact ly one root in the segment [a, b] which is m o r e - ove r s imple. If~ = x + #, then for sufficiently smal l Ifz[ uniformly with r e spec t to ] ([[]~ ----- l) we have

Im (L (£)],/) = Im ~ (L' (x) ], ]) + 0 ([ ~t I'), and this means that there exists a neighborhood U(C V) of the segment [a, b], such that (L (X) 1, /) ~ 0

,(1 =l = O, X ~ U \ [a, bl).

The asser t ion of the coro l la ry therefore follows f rom the theorem (it is necessa ry only to note that LO,) is invert ible for X 6 [a, 0) and X ~ (~A ~, b]).

For the case in which B(1) is a polynomial Coro l la ry 2 was obtained by H. Langer [2].

We note fur ther that if in Corol lary 2 the opera to r A is completely continuous, then by vir tue of (2) Z is also completely continuous, and the re fore the sys tem of e igenvectors of L(D corresponding to e igen- values in [0, |A ~] forms an unconditional basis in IL (see Corol lary 1). This asse r t ion under additional res t r ic t ions on A and BOO was obtained by Tu rne r [4] (see also [5]).

2. The following remarks contain ce r t a insupp lemen t s to the formulat ion of the theorem.

Remark 1. The opera to r Z, the existence of which is asser ted in the theorem, is uniquely d e t e r - mined.

Indeed, suppose that in U there is the equality L+ (X) (Z -- M)= I.+ (~.) ( ~ - M), where or(Z) c U, and L+(I) and Le(k) are holomorphic and invert ible in U. Then

(Z - - ~ / ) (Z - - ~.I) -~ = ~+x (X) L+ (~),

whereby the left side i sho lomorph ic outside a(Z) and equal to I for ~ = ~, while the right side is holomor- phic inUo T h e r e f o r e , ( Z - M ) ( Z - M ) -1 =- I, whence Z = Z.

For brevi ty everywhere in the sequel we understand by a contour a sufficiently smooth, simple closed + curve. If F is an a rb i t r a ry contour, then by r and F - we denote the connected components of the com- plement to F; it is hereby assumed that ~o E F- .

It obviously follows f rom the theorem that L(D is invertible in U \ [a; b].

Remark 2. Let F c U be an a r b i t r a r y contour enclosing [a, b]. F rom the proof of the theorem it will be evident that the opera tor

t i L-1 (k) d~, S = ~ 7 -

is uniformly positive and symmet r i ze s the opera tor Z f rom the right, i .e. , (ZS)* = ZS. Obviously, the opera tor Q = S - I is also uniformly positive and symmet r i ze s the opera to r Z f rom the left, i .e., (QZ)* = QZ (see also Lemma 6 where the explicit express ion (25) is given for the opera tor Q).

Remark 3. An opera to r function L(~), sat isfying the conditions of the theorem has the following proper ty (which in our view is very basic) . Let us consider an a rb i t r a ry complex number z t¢ [a, b] and choose a contour Y symmet r i c with respec t to the real axis such that z E F- . We set

R~ = ~:~- ~-y-Z7---"

Obviously, R z is a holomorphic operator function on C\ [a, b]. It is natural to consider R z eipal part of L-I(X). It turns out that

IR~ - (R~)'I >~ 0 (Ira z > o). 2~

Indeed,

the pr in-

I i L-I (k) iX -- z)(X-- } ) I r a z dL z, [Rz - - (Rz)'l = F

and since the opera to r function (Ira z)-lO, - z)(~ -z---)L(~) together with L(X) sat isf ies the conditions of the theorem (3) follows f rom Remark 2.

Remark 4. It follows f rom Eq. (1) that

def ~ , L (Z) = -- ~ f L (L) (Z -- LI) -~ d~. = 0 ([a, b] ~ F +, r c U).

r

If, in par t icu la r , L(h) is a polynomial of degree n, then this equation means that Z is a root of the opera tor equation of degree n L(Z) = 0 (see [1-3]).

(3)

§ 2 . A u x i l i a r y P r o p o s i t i o n s

1. Let B(X) be a holomorphic opera to r function in the domain G, and let / ~ @ (/=/= 0). We denote by N(B,f ) the set of roots of the function (B(~) f , ] ) in the domain G and set N(B) = U N(B,/).

Obviously, if B(/~) = A - M(A 6 ~), then N(B) = W(A), where W(A) is the numerical range of A, Joe.,

w (A) = {(Af, /3:111 II = i }

LEMMA 1. Let B(X) be a holomorphic opera tor function in the domain G. If 0 ¢ W(B(~0)) for some ~0 E G, then the spec t rum of B(D is contained in N(B).*

Proof . Ifta is a point of the spec t rum of B(D, then 0 ~ a(B(/~)), and hence 0 E W(B(u)), i .e. , there ex- ists a sequence {/~} C @(]l/~ll = l), such that (B(g) fn , Jn) ~ 0. Pass ing, if necessa ry , to a subsequence, it may be assumed that ( B ( D f n , f n ) converges in the domain G to a holomorphic function q~(D. Obviously,

*This iemma may be considered as a general izat ion of a well-known theorem to the effect that a(A) c W(A)

(A ~ ) .

~(~) = 0 and ~P(~0) ~ 0. By Hurwi tz ' t h e o r e m (see, fo r example , [6], p. 426) the re ex i s t s a sequence Pn ~ , fo r which ( B ( ~ n ) / n , f n ) = 0, and this comple t e s the p roof of the [ emma .

E v e r y w h e r e below we shal l a s s u m e the s e g m e n t In, b] to be fixed. F o r convenience we shal l s ay that any o p e r a t o r function sa t i s fy ing the condi t ions of the t h e o r e m is s imple in U. More p r e c i s e l y , an o p e r a t o r function L()0, which is ho lomorph ic and se l f - ad jo in t in s o m e s imply connected neighborhood U of the s e g - m e n t [a, b] we call s imp le inU if L(a) << 0, L(b) >> 0, and for a n y j ;~ 0 the funct ion ( L ( ~ ) f , f ) has exac t ly one root in U which is m o r e o v e r s imple . Obvious ly , this roo t l ies on the in te rva l (a, b).

F u r t h e r , until the end of the sec t ion L(~) denotes any o p e r a t o r funct ion which is s imp le in U. By v i r - tue of L e m m a 1 it is obvious that L(D is inver t ib le in U \ [ a , b].

LEMMA 2. There ex i s t num be r s ~, 5 > 0, such that the inequal i ty 1 (L (~0)/0, ]0) I <: e (~-o ~ [a, hi, ]lfoI[ : 1) impl ies the inequal i ty (L" (~0) ]o, ]o) ~ 6.

P roo f . If the a s s e r t i o n of the [ e m m a is not t rue , then the re ex is t sequences {/,,} C ~ (]f~[ = i) and {x.} C In, b], such that (L (z.) f . , f.) --~ 0, (L" (x.) f . , ].) -~ ~l~.~ 0.

P a s s i n g , if n e c e s s a r y , to a subsequence it m a y be a s sumed that x n -* x 0 and ( L ( h ) f n , ] n) conve rges u n i f o r m l y to some ho lomorph ic funct ion ~(~) in a neighborhood of the s e g m e n t In, b]. Then ~ (xo) = 0, ~' (xo) ~ 0, (p (b) ~ 0 . T h e r e f o r e , q~ 0.) has no fewer than two roo t s (counting mult ipl ici ty) on the s e g m e n t [a, b]. T h e r e f o r e , by Hurwi tz ' t h e o r e m ( L 0 0 f n , f n) has for suff ic ient ly l a rge n no fewer than two roots in U, which con t rad ic t s the s impl i c i ty of L ( D . This comple t e s the p roo f of the L e m m a .

2. We fix a E [a, b] such that the inequal i ty L(x) << 0 is t rue fo r al l x E [a, a ] and s e t

Lk(~) =e -~x L (k ) , Mt(~.) : L ~ ( k ) - - tL:~((z) (k ~ O , t ~ [ O , i]). (4)

LEMMA 3. The n u m b e r k > 0 can be chosen such that all the o p e r a t o r functions Mt(D (0 -< t <- 1) a r e s imp le in some neighborhood U of the s e g m e n t [a, b].

P roo f . F o r a n y f ~ ~ w e s e t

l/ (X) = (L (£) /, ]), lk, t (z) = (L~ (~) /, ]), mt,/ (£) (Mt (~) f, /). (5)

We shal l obtain a lower e s t ima te f o r the funct ion l~,t (x) = e -t~ [l i (x) -- kl I (x)] (x ~ [a, b], [I/It = 1).

By L e m m a 2 the re ex i s t e0, 50 > 0, such that l~ (x) ~ 6o, if t lt(x)[ < 8 0 a n d x E [ a , b ] . Le t k = 2so 1 max ~L'(x)~. Then fo r l t (x ) ~ - - eo we have l~..~ (x) ~ 1/~ keoe-~; if e0 <: l/(x) ~ 0, then l~.~ (x) ~ 5oe -~.

a~x~b

The o p e r a t o r function Lk(D is obv ious ly s imp le in U, and by L e m m a 2 the re ex is t ~ , 5 > 0 such that l~,~ ( x ) ~ t if 0 ~ l~,! ( x ) < e . Thus , i f l~,~ ( x ) ~ e , then l~,! (x) ~ 0 , w h e r e ~/ = rain {~/,keo e-~, ~oe -~, ~}.

F r o m this i t fol lows, in p a r t i c u l a r , tha t L~(x) -> 7 I fo r a ~ x ~ a. T h e r e f o r e , Ml(a) = Lk(a) - Lk(~) << 0, and hence Mr(a) << 0 fo r 0 -< t --< 1. C l ea r l y Mt(b) >> 0 (0 <-- t <- 1).

F u r t h e r , if me,! (x) < e (x ~ [a, b], II/ll = 1), then l~,~ (x) -- mr,! ( x )+ tl~,! (a) ~ e and hence m~,t (x) = l~,t (x) ~ ~ ~ 0. T h e r e f o r e the funct ion mt , f (x ) has exac t ly one roo t in the s e g m e n t [a, b] which is m o r e - o v e r s imple .

To comple te the p r o o f o f the l e m m a it r e m a i n s to choose a complex neighborhood U'of the s egmen t [a, b] such that m,,~ (~.) ~ 0 (X ~ ~1 \ [a, b], I =/= 0, 0 ~ t ~ i).

I f x ~ [a, b] and ?, = x + ~ , t h e n fo r sma l l I ~ [ u n i f o r m l y with r e s p e c t to f (l[/lI = t) and t ~ [0, 1]

Im rn~,~ (~) = I m ~ rni,/(x) + 0 (I 9 I~), Re mt,] (~) = m~,~(x) + 0 (]9[) .

F r o m this the re follows im m ed ia t e ly by L e m m a 2 the ex i s tence o f the r equ i red neighborhood of the s e g m e n t In, b]. This comple t e s the p r o o f of the l e m m a .

3. Le t L~(I-, (~) be the Hi lbe r t space of s t rong ly m e a s u r a b l e v e c t o r functions defined on F with v a l - ue's in 0, whe reby the s c a l a r p r o d u c t in L~(F, ~ ) is given by

(% ¢)~,(r, L~ (r, ~)). r

If )'0 ~ r + , then by L +~ (F, ~) (L~ (F, ~)) we denote the subspace of L~(F, ~), cons i s t ing of those v e c t o r funct ions ~p(p), fo r which

4

I (~ - - %~)~ q) (9) dl~ ---- 0 for n >~ 0 (n < 0).

I t is known that L2(F , ~) d e c o m p o s e s in the d i r e c t s u m of L~(F, @) and L ~ ( I , ~). By P we denote the o p e r a t o r of p ro j ec t ion f r o m L2(F, ~) onto L ~ ( Y , ~) p a r a l l e l to L~(F, ~).

For the v e c t o r funct ion q~ ~ L~ (r, 9) we denote by ~ (~) the fol lowing v e c t o r function which is ho lo-

morph ic in F+ :

(~) = 2~ ~ ~, - ~ (x ~ r+) (6)

~f B(X) is an operator function holomorphic in F + iJ I -~ and (p ~ L +~ (F, @), then, as is known,

B (~) ~ (ix) ~ L~ (r, @) (7)

and

~_k_t "(~) ~ (~) d~ ~ (~) ~ (~) (~ ~ r+) 2~i J ~--~ = (8)

P

We shal l need the following l e m m a which follows f r o m a gene ra l r e su l t of I. Ts . Gokhberg and Yu.

L a i t e r e r (see [7], T h e o r e m III . 1.1).

LEMMA 4. Le t A(F) be a s e l f - ad jo in t o p e r a t o r funct ion which is ho lomorph i c and inver t ib le on s o m e con tour I-, which is s y m m e t r i c with r e s p e c t to the r ea l ax i s . A(p) admi t s the r e p r e s e n t a t i o n

A (~) = A t (~) A_ (~) (~ ~ r), (9)

whe re the o p e r a t o r funct ion A~(g) is ho lom orph i c and inver t ib le on F ± U F and A_(~) = I if and only if the o p e r a t o r A defined in L2(F, ~) by the equat ion

A(p = P (A (~t) ~ (ix)), (10)

is inver t ib le .*

We shal l see below that Eq. (1) is a s imple consequence of the fac tored Eq. (9) f o r s o m e o p e r a t o r funct ion A(#), and we will t he r e fo re be requ i red to ve r i fy the inver t ib i l i ty of the c o r r e s p o n d i n g o p e r a t o r A. An impor t an t s tep in this d i r ec t ion is the p roof of the fac t that the o p e r a t o r A is left inver t ib le .

LEMMA 5. Le t F ( c U) be a con tou r enc los ing the in te rva l [a, b], let ~o ~ F+, and let A (F) = (~ -- )%)=1 L (~). Then the o p e r a t o r A defined by Eq. (10) is left inver t ib le .

P roo f . We shal l find an expl ic i t e x p r e s s i o n for the v e c t o r funct ion A~ (~ ~ L~ (F, ~)). It is obvious

that

A (,tt) (p (u) = L (~) ~ (~)-- L (~) ~ (~n) + L (~o) ~ (~o) (11)

The f i r s t t e r m on the r igh t s ide o f (11) be longs to L~ (F, ~), s ince for n >- 0

[L (ix) q~ (~) -- L (k~) T (k~)] (,~ - - k~) '~-~ dix : S L (~) (p (ix) (~ - - ~.~)~-~ d~ - - 2~i6,~oL (k~) ~ (k0) : 0 r P

[for n > 0 the las t equal i ty follows f r o m (7), while fo r n = 0 it follows f r o m (8)]. On the o the r hand, the second t e r m on the r igh t s ide of (11) obvious ly belongs to L~(1-, ~), and t h e r e f o r e

(A¢) (~t) = (~ -- ~0) -I (L (9)¢ (g) -- L (X0)~ (~'0)). (12)

Let us suppose that the o p e r a t o r A is not left inver t ib le . Then the re ex is t s a sequence {~=} c L~ (F, ~)

such that

'i~.llL i = i (n = 1, 2, 3 . . . . ) (13)

*In [7] the c o n c e r n is not with a f ac to r i za t ion of the f o r m (9) ("left f ac to r i za t ion" ) but r a t h e r with a f a c t o r - i za t ion of the f o r m A(~) = A_(~)A+(~) ( " r igh t f ac to r i za t ion" ) , but this is not i m p o r t a n t h e r e in view of the

s e l f - ad jo in tne s s o f A(~) and the s y m m e t r y of F .

and

F r o m (12) and (14) it fol lows that

We se t h~ = L ()'o)~, (~0).

l imlIL(IX)%~(~)--L(X3)~,~(X.~)t = 0. (15) IlL+

Since by L e m m a 1, L(~0 is inver t ib le away f rom [a, b] , it follows that

II h= H > c (rain [I L-~ (I x) ][-x il q~n ]lc~ -- It L (IX) T,, (IX) -- h~ IIL~ ) } ~ r

and because of (13) and (15) lim [Ih=ll > 0.

Le t ~n (~) = L (ix)q~ (~) -- hn. It follows f r o m (8) that {~ (~.) = L (~) ~ (~)-- h, (~ ~ r+), and t h e r e f o r e by (15)

L ()~) ~,~ ()~) ---- hn + o (1) for n --+ co (16)

un i fo rmly on each compac t s e t belonging to F +.

Le t V and V 1 be s imp ly connected ne ighborhoods of [a, b] which a r e s y m m e t r i c with r e s p e c t to the r ea l axis such that F, C V C V C F+. We denote V \ ~ 1 by U0. Since LO,) is inver t ib le in U 0, i t fol lows f r o m (16) that

~(~ . ) = L -x(~)h~ + o ( t ) (~.~_ Uo), (17)

whe re the r e m a i n d e r t e r m tends to z e r o un i fo rmly in U0 for n ~ ~o. F u r t h e r , we have

(9.(~), h.) = (L -~(~)h", h.) + o ( 1 ) ( X ~ Uo). (18)

Since by (16) h. = L ~) (~. (~) ÷ o (1), i t follows f r o m (18) that

(~.(~.), L ~ ) ~ n ( ~ ) ) = (L -~()~)h., h.) + o ( 1 ) ( ~ V 0 ) . (19)

The fami ly (L ()~) ~. ()~), ~ (~)) is un i fo rmly bounded in U 0 (s ince IIcP=[IL~ = t ). Pa s s ing to a s u b s e - quence , i t can be supposed that (L 0~) ~ 0~), ~ (~)) conve rges un i fo rmly in U0, and hence also in V, to some function ¢(X) which is' ho lomorph ic in V.

Dif ferent ia t ing Eq. (19) and pass ing to the l imi t , we find on cons ide r ing (17) that

¢ ' (~.) = - - l ira (L' ( ~ ) L "~ (~,~h,~, L -~ (~) h~) = - - lira (L' (~)~p. (~.), ~,, ~)). (20)

This equat ion is sa t i s f ied e v e r y w h e r e in U0 and hence also e v e r y w h e r e in V. Obviously , • (a) -.< 0, (b) ~> 0, and t h e r e f o r e the funct ion ¢ (D has roots on the s e g m e n t [a, b] . If x0 is the s m a l l e s t of these

roo t s , fl~en obvious ly

It follows f r o m (16) that

• ' (xo) ~ 0. (21)

limll~,,(x,,)il>~llL(x~)F' lim~h~ -{- o(1)l I = q > 0 , (22)

and s ince (L (xo) ~n (xo), ~ (xo)) --~ 0, i t follows f rom L e m m a 2 that

(L" (xJ~(xo), q). (xo)) > 8 II ~n (x0)ll ~ (n > no). (23)

F r o m (20), (23) and (22) we obtain the inequal i ty ~ ' (x 0) -< - 6q 2, which con t r ad ic t s (21). This c o m - p le tes the p r o o f of the l e m m a .

4. The next l emma is due to V. I. L o m o n o s o v .

L E M M A 6 . L e t B(~ ) be a se l f - ad jo in t o p e r a t o r function in a s imply connected domain G s y m m e t r i c with r e s p e c t to the r e a l axis which admi ts the r e p r e s e n t a t i o n

B (~.) ---- B+ (~.) (Z - - ~./) (~. ~ G), (24)

where B+(X) is holomorphic and invertible in G and or(Z) 6 G. If r (~ G) is an a rb i t r a ry contour enclosing a (Z), then the opera tors

S = +iB-~(~ , )d~ , , Q = ~ I (Z ' -~d) ' 'B (~ ' ) (Z-~ ' l ) -~d)~ (25)

are setf-adjoint and inverse to one another.

Proof. Since B(X) is invertible in G\a (Z) , it follows that S and Q do not depend on which contour be- longing to G and enclosing a(Z) we choose. F rom the self-adjointness of the function B(X) it follows that or(Z) is symmet r i c with respec t to the real axis. Therefore , there exist contours r t , r 2 ~ G , symmet r i c with respec t to the real axis such that r~ ~ r~, z (z) ~ r~ and

S = B -1 (~) d~, 0 = ~ (Z* -- ~I) -~ B (~) (Z -- ~I) "~ dk. r t

Obviously, S and Q are self-adjoint opera tors and

QS = ~ t f I (Z* -- ~I) -~ B (X) (Z - - ~,i)-1 (Z -- ~I)-' B~ ~ (~) @ dk. r~ r ,

From the Hilbert identity

+ Since X E r 2 , it follows that

B.7.1 (l~) d~t~ d~.

P t r ,

'S :}" = - ~ (z" - x[)-' B+ (X) B; ' 0,) d~ = -- ~ - (Z -- X[)-~ dX = L Fx

F r o m the self-adjointness of the opera tor function B(X) f rom Eq. (24) it follows easily that B(M = (Z* - XI) [B+(~)]*, and therefore

2 t 2 = ~ \,) ~_~

~+ + Since t~ ~ I~ U I~, in the inner integral the integrand is holomorphic o n L 1 I J L1 , and hence 2t2 = O.

Thus, QS -- I, and by virtue of the self-adjointness of the opera tors Q and S we find that SQ = I. This eom- pletes the proof of the lemma.

it

We remark that in the case where B (~) = ~ ~JAj, the opera tor Q admits also the following s impler

representat ion Q = ~ ~ (Z*)~A~+~Z ~-~ (see [1, 3]). ~ = o Y = o

§ 3 . P r o o f o f t h e T h e o r e m

Let k and ~ be chosen in cor respondence with Lemma 3, and let L = ~ b e a contour enclosing [a, b]. We set

At (p.) = ( p ~ - - a) -1 Mt (~t), At ¢p = P (At (~) q~ (~)) (~0 ~ L~(F, ~)).

It follows f rom Lemmas 3 and 5 that for all t E [0, 1] the opera tor A t is left invertible. I n ' w e con- s ide r the simple function MI(M [see (4)]. Obviously, Ml(a) = 0, and therefore the function (~-- a) -1 ml, I(X) [where me r(),) is defined in (5)] has no roots in ~ and hence by Lemma 1 the holomorphic opera tor func-

v , d 1 " " tion (), - ~r)- MI(X) is revertible everywhere in U. +

Thus, the opera tor A 1 is invertible in L 2 ( r , ~), as the opera tor of multiplication by the function Al(g) which is holomorphic and invertible in a neighborhood of Y+ U F [see (7)]. Therefore , f rom the left in- vert ibi l i ty of the opera tors At and the continuity of A t as functions of t (in the opera tor norm) it follows that all the opera tors At are invertible (see, for example, [8], p. 202). In par t icu lar , the opera tor A0 is in- vert ible. Applying L e m m a 4, we obtain a factored equation for the function A 0 (1~)= (~t- ~)-1L~ (~)

(~t -- a) -i Lk (~) = L~ (~t)L;(~) (~t ~ r)

o r

L ; (~t) = (t~ - - a ) - l i L ~ • (~)]-'L~ (~t) (~ ~ F).

The o p e r a t o r function Lk00 is holomorphic and inver t ible on r - U F and equal to I for h = ~ , w h i l e ()~ -- a) -1 [L~ ()~)]-~ L~ ()~) is holomorphic on F + U F with the exception of the point ~ where it has a s imple pole. Hence L~(M has the f o r m

X L~()~)---- I T )~_~ ( X ~ ) ,

and therefore

L (~t) ---- L+ (~t) (Z - - ~tl) (Z = aI -- X, p ~ F), (26)

where the operator function L+ (~) = -- ek~L~ ()~) is holomorphic and invertible on Y+ U F, and a(Z) c F +. Since by Lemma 1, L(~) is invertible {n U\[a, b], it follows from (26) that a(Z) c [a, b], while L+00 is holomorphic and invertible everywhere in U.

I t thus r emains only to prove that the ope ra to r Z is s i m i l a r to a se l f -ad jo in t ope ra to r . We se t Ct =

t IM[~(~,)d~ ' ([a, blr_r+, r ~ 7 ) P

According to L e m m a 3 the ope ra to r functions Mt(~)(0 -< t --< 1) a re s imple i n ~ , and there fore the pa r t of the theorem which has been proved is valid for them, i .e . ,

M , (X) = M : ()~) (Z, - - ~,I),

where M~().) is holomorphic and invert ible in ~ and J(Zt) ~ [a, b]° By L e m m a 6 it follows f r o m this that the ope ra to r s Ct(0 -< t -< 1) a re se l f -ad jo in t and inver t ib le .

Since MI(~) = 0, by L e m m a 2 M~(a) >> 0, and s ince the ope ra to r function (~ - o~)-lMl(),) is ho lomor - phic and inver t ible in U, it follows that

Mi 1 ()~) ---- (~, -- a) -i [M~ (a)]-i + F (~.),

where F(M is holomorphic in ~. Thus, C 1 = [M~(a)] - I (>> O), and inasmuch as Ct depends continuously on t in ope ra to r no rm C t >> 0 for all t e [0, 1]. We se t

P

Recall ing that Sk(= Co) >> 0 and arguing as above, we find that S o >> O. It is eas i ly seen that

r r v r

and since the contour r can be a s sumed s y m m e t r i c with r e spec t to the rea l axis it follows that (ZS0)* = ZS 0. I t r ema ins to note that Z = S~/2 HS~ 1/~, where H = S o " (ZS~)S~ v' ( = / / ' ) . This comple tes the proof of the theo rem.

I.

21

3.

4.

5.

6.

L I T E R A T U R E C I T E D

M. G. Krein and H. Langer , "On ce r t a in ma themat i ca l pr inc ip les of the l inear theory of damped o s c i l - lations of continua," Proceedings of the Internat ional Symposium on Applications of Function Theory in the Mechanics of Continuous Media, Vol. 2, Nauka, Moscow (1965), pp. 284-322. H. Langer "Uoer eine Klasse n ich t l inea re r E igenvalueprobleme," Acta Scient. Math., 3__55, 79-93 (1973}. A. S. Markus , V. I. Matsaev, and G. I. Russu, "On ce r ta in genera l iza t ions of the theory of s t rongly damped pencils to the case of penci ls of a r b i t r a r y o r d e r , " Acta Scient. Math., 34, 245-271 (1973)' R. E. L. Tu rne r , "A c lass of nonl inear eigenvalue p r o b l e m s , " J . Funct. Anal. , 2, No. 3 ,297-322 (1968). A. S.Markus and G. I. Russu, "On a basis formed from the eigenvectors of a self-adjoint polynomial pencil," Matem. Issledovaniyai 6, No. 1,114-125 (1971). A. I. Markushevich, Theory of Analytic Functions [in Russian], Vol. 1, Nauka, Moscow (1967).

7o

8.

I. Ts° Gokhberg and Yu. L a i t e r e r , "Genera l t heo rems on the canonical fac tor iza t ion of ope ra to r functions with r e s pec t to a contour ," Matem. Iss ledovaniya 7, No. 3, 87-134 (1972). I. Ts . Gokhberg and M. G. Krein, The Theory of Vo l t e r r a Ope ra to r s in Hi lber t Space and Its Appli- cations [in Russian] , Nauka, Moscow (1967).