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Remarks on the spectrum of a compact convex setof compact operators Ky Anh Pham and Xuan Thao Nguyen Abstract.. In this note we derive a necessary and sufficient condition for a compact

Remarks on the spectrum of a compact convex set

of compact operators

Ky Anh Pham and Xuan Thao Nguyen

Abstract In this note we derive a necessary and sufficient condition for a compact convex

set of linear compact operators acting in a complex Hilbert space to have the spectrum

outside a prescribed closed convex subset of the complex plane

Keywords Compact operator, extreme points, perturbation of the spectrum.

2010 Mathematics Subject Classification 47A10, 47A25, 47A55, 47A75.

Recently, an interesting result characterizing some spectral properties of a

com-pact convex set of matrices in terms of its extreme points has been obtained by

Monov [6] His result generalizes known criteria for the stability and

nonsingular-ity of all matrices belonging to a compact convex set of matrices [1, 2]

Monov’s result relies essentially on the continuous dependence of the spectrum

of a matrix on its entries, which is no longer true for bounded linear operators

The aim of this note is to show that Monov’s result remains valid for compact

operators in infinite-dimensional complex Hilbert spaces

Denote by B.H / and K.H / the spaces of all bounded linear operators and the

set of all compact operators acting in H; respectively Then, K.H / is a closed

subspace of B.H /:

Let M be a convex compact subset of K.H /: We define the spectrum of the set

M as

 M/ D ¹ 2  A/ W A 2 Mº:

Since M  K.H /; the spectrum  M/ always contains the origin, i.e., 0 2  M/:

Let ƒ be a closed convex subset in C: In this paper we will derive a necessary

and sufficient condition for the spectrum  M/ to lie outside ƒ, i.e.,

The following facts about the approximation and seperation properties of closed

convex sets in Hilbert spaces can be found in many books on optimization and

ill-posed problems, for example, see [3, 7]

Lemma 1 Let S  H be a nonempty closed and convex set and x 2 H be an

arbitrary fixed point Then the following assertions hold:

(i) There is a unique point y02 S; such that

(ii) if x … S; then the relation (2) holds if and only if Rehy0 x; y  y0i  0,

8y 2 S:

Recall that a point x of a convex set S is called extreme, if there do not exist

y; z 2 S and  2 0; 1/; such that x D y C 1  /z: The closed convex hull of

an arbitrary set M in a locally convex space is denoted by Co M: In what follows,

we will denote by E the set of all extreme points of M  K.H /:

Lemma 2 For any fixed points x; y 2 H; the following relation holds:

sup A2M

RehAx; yi D sup

A2E RehAx; yi:

Proof According to Krein–Milman’s theorem on extreme points, see [5, p 46],

M D Co E: Hence, d WD supA2MRehAx; yi D supA2Co ERehAx; yi  d WD

supA2ERehAx; yi: On the other hand, 8  0; 9n D n./ 2 N; 9Ai 2 E, 9˛i 

0 (i D 1; : : : ; n/ such thatPn

i D1˛i D 1 and d    Reh.Pn

i D1˛iAi/x; yi D

Pn

i D1˛iRehAix; yi  Pn

i D1˛i/d D d  d: Thus, d D d ; which was to be proved

Let ƒ0D ƒ n @ƒ: It would be useful to mention that ƒ0is the interior of ƒ:

Lemma 3 Suppose that 0 … ƒ and the boundary @ƒ is a rectifiable simple closed

curve Moreover, assume that

and there is an operatorA12 M; such that

Then the relation (1) holds.

Proof We prove the lemma by supposing a contradiction, that  M/ \ ƒ ¤ ;;

i.e., there exists A02 M; such that  A0/ \ ƒ ¤ ;: From (3), it follows

Consider the compact operator A˛ WD ˛A1C 1  ˛/A0for ˛ 2 Œ0; 1: Conditions

(3), (4) imply  A1/  C n ƒ: Using the compactness of the spectrum  A˛/ and

the openness of C n ƒ as well as taking into account the upper-semicontinuity of

the map ˛ !  A˛/ (see [4, Chap IV, Remark 3.3]), we have  A˛/  C n ƒ for

all ˛ sufficiently close to 1

Define ˛ WD inf¹˛ 2 Œ0; 1 W  Aˇ/  C n ƒ, 8ˇ 2 ˛; 1º: If  A˛/  C n ƒ

again, then the minimality of ˛ ensures ˛ D 0; which contradicts condition (5)

Thus  A˛/ \ ƒ ¤ ;: Taking into account the relation (3), we find that ˛ WD

 A˛/ \ ƒ0¤ ;: Clearly, the spectrum of the compact operator A˛ is seperated

into 2 parts ˛and ˛0 WD  A˛/ n ˛ by the rectifiable simple closed curve @ƒ:

Moreover, ˛ consists of a finite number of eigenvalues of A˛; otherwise, it has

an accumulation point  D 0 … ƒ; which is impossible The openness of ƒ0

and the finiteness of ˛ imply that ˛./ WD ¹ 2 C W d.; ˛/ < º  ƒ0for

a sufficiently small : According to Kato [4, p 213], the part of the spectrum ˛

changes with A˛; and hence, with ˛; continuously Hence, there is ˛0 2 ˛; 1

sufficiently close to ˛; such that  A˛0/ has a part ˛ 0  ˛./  ƒ0; which

implies  A˛ 0/ \ ƒ0 ¤ ;: On the other hand, by the definition of ˛;  A˛/ 

Cn ƒ, 8˛ 2 ˛; 1: The obtained contradiction proves the lemma

Remark 1. a According to Kato [4, p 213], the requirement that @ƒ is a

recti-fiable simple closed curve is essential for the continuity of ˛with respect to

the parameter ˛:

b Lemma 3 holds trivially if ƒ is a convex closed subset, such that ƒ  @ƒ;

i.e., when ƒ is a straight line or a segment In this case, the interior of ƒ is

empty, hence (4) is satisfied trivially and relations (3) and (1) are equivalent

Now we are able to state the main result, whose proof can be carried out

simi-larly as in [6]

Theorem 1 Let ƒ  C be a closed convex subset and 0 … ƒ: Further, suppose

that M  K.H / is a convex compact set Then, a necessary and sufficient

condi-tion for the equality (1) is the following requirements:

(i) There is an operator A 2 M; such that

(ii) 8x 2 H n ¹0º, 9y D y.x/ 2 H :

inf

2@ƒRehx; yi > sup

A2E

Proof Necessity Let (1) be satisfied Then, relation (6) holds automatically For

an arbitrary fixed x ¤ 0 we consider two closed convex subsets S1 D ¹x W

 2 ƒº and S2 D ¹Ax W A 2 Mº: Then, the difference S D S1  S2 D

¹y D x  Ax W  2 ƒI A 2 Mº is again a closed and convex set

Evi-dently, 0 … S; otherwise,  2  A/ \ ƒ   M/ \ ƒ; which contradicts (1)

By Lemma 1, there exists a unique point y0 2 S; such that 0 < ky0k  kyk,

8y 2 S: Moreover, Rehy0; y  y0i  0, 8y 2 S: The last inequality

im-plies Rehy0; xi  RehAx; y0i  ky0k2 > 0, 8 2 ƒ, 8A 2 M: Hence,

inf2ƒRehx; y0i > supA2MRehAx; y0i: By Lemma 2, supA2MRehAx; y0i D

supA2ERehAx; y0i: Besides, inf2@ƒRehx; y0i  inf2ƒRehx; y0i: Thus the

relation (7) is proved

Sufficiency We show that condition (7) implies the relation (3) Indeed,

sup-pose by contradiction that there exist B 2 M;  2 @ƒ and some x 2 H n ¹0º; such

that Bx D x; or equivalently, hx; yi D hBx; yi8y 2 H: Using Lemma 2 and

taking into account the last equality, we get inf2@ƒRehx; yi  Rehx; yi D

RehBx; yi  supA2MRehAx; yi D supA2ERehAx; yi, 8y 2 H; which

contra-dicts (7) for some y D y.x/: Thus (3) is proved Now we consider three cases

(a) First let ƒ be a convex closed subset without an interior, i.e., ƒ  @ƒ By

the above mentioned arguments, condition (7) implies (3), which is equivalent to

(1) in this case

(b) Further, suppose that ƒ is a closed convex and bounded subset with a

non-empty interior Then @ƒ is a rectifiable simple closed curve (see, e.g., [8,

Prob-lem 1.5.1]) Taking into account conditions (3), which follows from (7), and (6),

and using Lemma 3 we come to the relation (1)

(c) Finally, let ƒ be a closed convex and unbounded subset with nonempty

interior Since M is compact, there is a positive number r such that kAk  r for

all A 2 M: Further, there exists a sufficiently large number n > r such that the set

ƒnD ¹ 2 ƒ W jj  nº is nonempty Observing that

 M/  Sr WD ¹ W jj  rº; (8)

we come to the conclusion that  M/ \ ƒ D ; if and only if  M/ \ ƒn D ;:

The set ƒn is a closed convex and bounded set with a rectifiable simple closed

boundary @ƒn D L1;n[ L2;n; where L1;n D ¹ 2 @ƒ W jj  nº and L2;n D

¹ 2 ƒ0 W jj D nº: Clearly, 0 … ƒn and  M/ \ @ƒn D  M/ \ L1;n/ [

. M/ \ L2;n/ D ;: Thus, all the conditions of Lemma 3 for ƒnare satisfied,

hence  M/ \ ƒn D ;; which was to be proved

We end this paper by considering some illustrative examples

Example 1 Let Ai; i D 1; 2; 3 be compact operators in `2 given by

infinite-dimensional block diagonal matrices Ai D diag.A1i; A2i; : : : ; Ani; : : : /, where

each Ani is a two by two matrix, whose all entries are .n C i /1; n D 1; 2; : : : ;

i D 1; 2; 3: Let ƒ D ¹z 2 C W z D a C i b; a  1; b 2 Rº and M be a convex hull

of the operators Ai, i D 1; 2; 3: Then E D ¹A1; A2; A3º: Obviously, condition (6)

holds for each Ai; while condition (7) is fulfilled for any x ¤ 0 and y D x:

In-deed, we have supA2ERehAx; yi D P1

nD1.n C 3/1jx2n1C x2nj2  0 <

kxk2D infb2RRe.1 C bi /kxk2D inf2@ƒRehx; xi: According to Theorem 1,

the relation (1) holds

The following example shows that if the set ƒ is not convex, then Theorem 1 is

not true, i.e., relation (1) may be satisfied although (7) does not hold

Example 2 Let ƒ D ¹z 2 C W jzj  1º: Consider two compact operators A D O

and B D diag B1; B2; : : : ; Bn; : : : /; where each Bnis a two by two matrix with

all entries being equal to n C 2/1: Clearly, the spectrum of the segment ŒO; B

lies outside ƒ: However, condition (7) does not hold, since for any x ¤ 0 and for

any y we have inf2@ƒRehx; yi D inf 2Œ0;2eijhx; yij D jhx; yij  0: On

the other hand, supA2ERehAx; yi  RehOx; yi D 0:

In applications the set ƒ  C usually contains the origin, hence zero always

be-longs to  M/ \ ƒ whenever H is infinite-dimensional This is a main difference

of the infinite-dimensional case from the finite one

Acknowledgments The authors would like to express their special thanks to the

referees, whose careful reading and many constructive comments led to a

consid-erable improvement of the paper The work is partially supported by the Vietnam

National Foundation for Science and Technology Development (NAFOSTED)

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Received April 21, 2008; revised August 5, 2009

Author information

Ky Anh Pham, Department of Mathematics, Vietnam National University,

334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam

E-mail: anhpk@vnu.edu.vn

Xuan Thao Nguyen, Department of Mathematics, Vietnam National University,

334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam

E-mail: thaonx281082@yahoo.com