DSpace at VNU: A minimized assumption generation method for component-based software verification

Journal of Applied Analysis 16 (2010), 259 – 264 DOI 10.1515 / JAA.2010.017 © de Gruyter 2010 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 compact 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 nonsingularity 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 A/ W A Mº: Since M K.H /; the spectrum M/ always contains the origin, i.e., 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., M/ \ ƒ D ;: (1) 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] Brought to you by | New York University Bobst Library Technical Services Authenticated Download Date | 5/17/15 12:00 AM 260 K A Pham and X T Nguyen Lemma Let S H be a nonempty closed and convex set and x H be an arbitrary fixed point Then the following assertions hold: (i) There is a unique point y0 S; such that kx y0 k D d.x; S /I (ii) if x … S; then the relation (2) holds if and only if Rehy0 8y S: (2) x; y y0 i 0, Recall that a point x of a convex set S is called extreme, if there not exist y; z S and 0; 1/; such that x D y C /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 For any fixed points x; y H; the following relation holds: sup RehAx; yi D sup RehAx; yi: A2M A2E Proof According to Krein–Milman’s theorem on extreme points, see [5, p 46], M D Co E: Hence, d WD supA2M RehAx; yi D supA2Co E RehAx; yi d WD supA2E RehAx; yi: On the other hand, 0; 9n D n / N; 9A E, 9˛ i i P P Ä Reh niD1 ˛i Ai /x; yi D 0P(i D 1; : : : ; n/ such thatP niD1 ˛i D and d n n iD1 ˛i RehAi x; yi Ä iD1 ˛i /d D d Ä d: Thus, d D d ; which was to be proved Let ƒ0 D ƒ n @ƒ: It would be useful to mention that ƒ0 is the interior of ƒ: Lemma Suppose that … ƒ and the boundary @ƒ is a rectifiable simple closed curve Moreover, assume that M/ \ @ƒ D ;; (3) and there is an operator A1 M; such that A1 / \ ƒ0 D ;: (4) Then the relation (1) holds Proof We prove the lemma by supposing a contradiction, that M/ \ Ô ;; i.e., there exists A0 M; such that A0 / \ ƒ ¤ ;: From (3), it follows A0 / \ ƒ0 ¤ ;: (5) Brought to you by | New York University Bobst Library Technical Services Authenticated Download Date | 5/17/15 12:00 AM The spectrum of a compact convex set of compact operators 261 Consider the compact operator A˛ WD ˛A1 C ˛/A0 for ˛ 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 Define ˛ WD inf¹˛ Œ0; 1 W Aˇ / C n ƒ, 8ˇ ˛; 1º: If A˛ / C n ƒ again, then the minimality of ˛ ensures ˛ D 0; which contradicts condition (5) Thus A / \ Ô ;: Taking into account the relation (3), we find that WD A / \ Ô ;: Clearly, the spectrum of the compact operator A˛ is seperated into 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 … ƒ; which is impossible The openness of ƒ0 and the finiteness of ˛ imply that ˛ / WD ¹ C W d ; ˛ / < º ƒ0 for a sufficiently small : According to Kato [4, p 213], the part of the spectrum ˛ changes with A˛ ; and hence, with ˛; continuously Hence, there is ˛ ˛; 1 sufficiently close to ˛; such that A˛0 / has a part ˛0 ƒ0 ; which ˛ / implies A˛0 / \ Ô ;: On the other hand, by the definition of ˛; A˛ / C n ƒ, 8˛ ˛; 1: The obtained contradiction proves the lemma Remark a According to Kato [4, p 213], the requirement that @ƒ is a rectifiable simple closed curve is essential for the continuity of ˛ with respect to the parameter ˛: b Lemma holds trivially if ƒ is a convex closed subset, such that ƒ Á @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 similarly as in [6] Theorem Let ƒ C be a closed convex subset and … ƒ: Further, suppose that M K.H / is a convex compact set Then, a necessary and sufficient condition for the equality (1) is the following requirements: (i) There is an operator A M; such that A/ \ ƒ0 D ;: (6) (ii) 8x H n ¹0º, 9y D y.x/ H : inf Reh x; yi > sup RehAx; yi: 2@ƒ (7) A2E Brought to you by | New York University Bobst Library Technical Services Authenticated Download Date | 5/17/15 12:00 AM 262 K A Pham and X T Nguyen Proof Necessity Let (1) be satisfied Then, relation (6) holds automatically For an arbitrary fixed x Ô we consider two closed convex subsets S1 D ¹ x W ƒº and S2 D ¹Ax W A Mº: Then, the difference S D S1 S2 D ¹y D x Ax W ƒI A Mº is again a closed and convex set Evidently, … S; otherwise, A/ \ ƒ M/ \ ƒ; which contradicts (1) By Lemma 1, there exists a unique point y0 S; such that < ky0 k Ä kyk, 8y S: Moreover, Rehy0 ; y y0 i 0, 8y S: The last inequality implies Rehy0 ; xi RehAx; y0 i ky0 k2 > 0, ƒ, 8A M: Hence, inf 2ƒ Reh x; y0 i > supA2M RehAx; y0 i: By Lemma 2, supA2M RehAx; y0 i D supA2E RehAx; y0 i: Besides, inf 2@ƒ Reh x; y0 i inf 2ƒ Reh x; y0 i: Thus the relation (7) is proved Sufficiency We show that condition (7) implies the relation (3) Indeed, suppose by contradiction that there exist B M; @ƒ and some x H n¹0º; such that Bx D x; or equivalently, h x; yi D hBx; yi8y H: Using Lemma and taking into account the last equality, we get inf 2@ƒ Reh x; yi Ä Reh x; yi D RehBx; yi Ä supA2M RehAx; yi D supA2E RehAx; yi, 8y H; which contradicts (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 Á @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 nonempty interior Then @ƒ is a rectifiable simple closed curve (see, e.g., [8, Problem 1.5.1]) Taking into account conditions (3), which follows from (7), and (6), and using Lemma 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 Ä r for all A M: Further, there exists a sufficiently large number n > r such that the set ƒn D ¹ ƒ W j j Ä nº is nonempty Observing that M/ Sr WD ¹ W j j Ä 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 @ƒ W j j Ä nº and L2;n D ¹ ƒ0 W j j D nº: Clearly, … ƒn and M/ \ @ƒn D M/ \ L1;n / [ M/ \ L2;n / D ;: Thus, all the conditions of Lemma for ƒn are satisfied, hence M/ \ ƒn D ;; which was to be proved We end this paper by considering some illustrative examples Example Let Ai ; i D 1; 2; be compact operators in `2 given by infinitedimensional block diagonal matrices Ai D diag.A1i ; A2i ; : : : ; Ani ; : : : /, where Brought to you by | New York University Bobst Library Technical Services Authenticated Download Date | 5/17/15 12:00 AM The spectrum of a compact convex set of compact operators 263 each Ani is a two by two matrix, whose all entries are n C i / ; n D 1; 2; : : : ; i D 1; 2; 3: Let ƒ D ¹z C W z D a C i b; a 1; b Rº and M be a convex hull of the operators Ai , i D 1; 2; 3: Then E D ¹A1 ; A2 ; A3 º: Obviously, condition (6) holds for each Ai ; while condition (7) isPfulfilled for any x Ô and y D x: In1 deed, we have supA2E RehAx; yi D nD1 n C 3/ jx2n C x2n j Ä < kxk2 D infb2R Re.1 C bi /kxk2 D inf 2@ƒ Reh x; xi: According to Theorem 1, the relation (1) holds The following example shows that if the set ƒ is not convex, then Theorem is not true, i.e., relation (1) may be satisfied although (7) does not hold Example Let ƒ D ¹z C W jzj 1º: Consider two compact operators A D O and B D diag B1 ; B2 ; : : : ; Bn ; : : : /; where each Bn is a two by two matrix with all entries being equal to n C 2/ : Clearly, the spectrum of the segment ŒO; B lies outside ƒ: However, condition (7) does not hold, since for any x Ô and for any y we have inf 2@ƒ Reh x; yi D infÂ2Œ0;2  e i jhx; yij D jhx; yij Ä 0: On the other hand, supA2E RehAx; yi RehOx; yi D 0: In applications the set ƒ C usually contains the origin, hence zero always belongs 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 considerable improvement of the paper The work is partially supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) Bibliography [1] S Bialas, A necessary and sufficient condition for the stability of convex combinations of stable polynomial and matrices, Bull Polish Acad Sci Tech Sci., 33 (1985), 473– 480 [2] N Cohen and I Lewkowicz, A necessary and sufficient criterion for the stability of a convex set of matrices IEEE Trans Automat Contr., 38 (1993) 611–615 [3] C W Groetsch, Inverse Problems in the Mathematical Sciences, Vieweg Verlag, Wiesbaden, 1993 [4] T Kato, Pertubation Theory for Linear Operators, Springer Verlag, Berlin, Heidelberg, New York, 1980 [5] S G Krein et al., Functional Analysis, Nauka, Moscow, 1972 (in Russian) Brought to you by | New York University Bobst Library Technical Services Authenticated Download Date | 5/17/15 12:00 AM 264 K A Pham and X T Nguyen [6] V V Monov, On the spectrum of convex sets of matrices, IEEE Trans Automat Contr., 44 (5)(1999), 1009–1012 [7] B N Pshenichny and Yu.M Danilin, Numerical Methods in Extremal Problems, Mir Publishers, Moscow, 1978 [8] V A Toponogov, Differential Geometry of Curves and Surfaces, Birkhauser, 2005 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 Brought to you by | New York University Bobst Library Technical Services Authenticated Download Date | 5/17/15 12:00 AM ... 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... Differential Geometry of Curves and Surfaces, Birkhauser, 2005 Received April 21, 2008; revised August 5, 2009 Author information Ky Anh Pham, Department of Mathematics, Vietnam National University,... Library Technical Services Authenticated Download Date | 5/17/15 12:00 AM The spectrum of a compact convex set of compact operators 263 each Ani is a two by two matrix, whose all entries are