Research Article Perturbations of Polynomials with Operator Coefficients

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1 Coplex Analysis Volue 2013, Article ID , 5 pages Research Article Perturbations of Polynoials with Operator Coefficients Michael Gil Departent of Matheatics, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva, Israel Correspondence should be addressed to Michael Gil ; gili@bezeqint.net Received 18 Deceber 2012; Accepted 20 February 2013 Acadeic Editor: Janne Heittokangas Copyright 2013 Michael Gil. This is an open access article distributed under the Creative Coons Attribution License, which perits unrestricted use, distribution, and reproduction in any ediu, provided the original work is properly cited. We consider polynoials whose coefficients are operators belonging to the Schatten-von Neuann ideals of copact operators in a Hilbert space. Bounds for the spectra of perturbed pencils are established. Applications to differential and difference equations are also discussed. 1. Introduction and Preliinaries Nuerous atheatical and physical probles lead to polynoial operator pencils (polynoials with operator coefficients); cf. [1 and references therein. Recently, the spectral theory of operator pencils attracts the attention of any atheaticians. In particular, in the paper [2, spectral properties of the quadratic operator pencil of Schrődinger operators on the whole real axis are studied. The author of the paper [3 establishes sufficient conditions for the finiteness of thediscretespectruoflinearpencils.thepaper[4 deals with the spectral analysis of a class of second-order indefinite nonself-adjoint differential operator pencils. In that paper, a ethod for solving the inverse spectral proble for the Schrődinger operator with coplex periodic potentials is proposed. In [5, 6, certain classes of analytic operator valued functions in a Hilbert space are studied, and bounds for the spectra of these functions are suggested. The results of papers [5, 6 are applied to second-order differential operators and functional differential equations. The paper [7 considers polynoial pencils whose coefficients are copact operators. Besides, inequalities for the sus of absolute values and real and iaginary parts of characteristic values are derived. The paper [8 is devoted to the variational theory of the spectra of operator pencils with self-adjoint operators. A Banach algebra associated with a linear operator pencil is explored in [9. A functional calculus generated by a quadratic operator pencil is investigated in [10. A quadratic pencil of differential operators with periodic generalized potential is considered in [11. The fold copleteness of a syste of root vectors of a syste of unbounded polynoial operator pencils in Banach spaces is explored in [12. Certainly, we could not survey the whole subject here and refer the reader to the pointed papers and references cited therein. Note that perturbations of pencils with nonself-adjoint operator coefficients, to the best of our knowledge, were not investigated in the available literature, although in any applications, for exaple, in nuerical atheatics and stability analysis, bounds for the spectra of perturbed pencils are very iportant; cf. [13. In the present paper, we derive such bounds in the case of polynoials whose coefficients are operators belonging to the Schattenvon Neuann ideals of copact operators in a Hilbert space. Introduce the notations. Let H be a separable coplex Hilbert space with a scalar product (, ) and the nor = (, ).ByI the identity operator in H is denoted. For a linear operator A in H, A 1 is the inverse operator, σ(a) is the spectru, λ k (A) (k = 1, 2,...) are the eigenvalues with their ultiplicities, A is the adjoint operator, A is the operator nor, and R λ (A) = (A λi) 1 is the resolvent. SN (1 < ) is the Schatten-von Neuann ideal of copact operators K in H having the finite nor N (K) = [Trace(KK ) /2 1/. Let A k and A k (k = 1,..., < )belinearbounded operators in H. Consider the pencils P (λ) = k=0 A k λ k, P (λ) = k=0 A k λ k ( A 0 =A 0 =I). (1)

2 2 Coplex Analysis Apointλ C is called a regular value of P,ifP (λ) is boundedly invertible. The copleent of all regular points of P to the closed coplex plane is called the spectru of P and is denoted by Σ(P ). Our ain proble is as follows: if A k and A k (k=1,...,) are close, how close are the spectra of P (λ) and P (λ)? For an integer j 1,wewillsaythat zsv (j) P ( P ):= sup μ Σ( P ) inf s Σ(P ) μj s j (2) is the j-spectral variation of P with respect to P.LetT= T(P ) and T = T( P ) be the operator -atrices definedonthetheorthogonalsuh of exeplars of H by A 1 A 2 A 1 A I T=( 0 I 0 0 ), 0 0 I 0 A 1 A 2 A 1 A T =( I I 0 0 ). 0 0 I 0 Lea 1. Let all the operators A k (k = 1,...,)belongto soe ideal J ofcopactoperators.then,alltheentriesofthe operator atrix T also belong to J. Proof. For an integer μ<, consider the operator atrix b 11 b 12 b 1, μ b 1, 1 b 1, b 21 b 22 b 2, μ b 2, 1 b 2, B,μ = b μ1 b μ,2 b μ, μ b μ, 1 b μ, ( I ) 0 I ( 0 0 I 0 0 ) with soe operators b jk. Direct calculations show that the operator atrix D,μ+1 := B,μ T has the for (3) (4) d 11 d 12 d 1, μ 1 d 1, 1 d 1, d 21 d 22 d 2, μ 1 d 2, 1 d 2, d μ1 d μ2 d μ, μ 1 d u, 1 d μ, D,μ+1 = d μ+1,1 d μ+1,2 d μ+1, μ 1 d μ+1, 1 d μ+1,, ( I ) 0 I (5) ( 0 0 I 0 0 ) c (μ+1) jμ = c (μ) j1 A (1 j μ), d jk = b j1 A k +b j,k+1 (1 j μ,k<), c (μ+1) μ+1,k = A k. (7) d j = b j1 A d μ+1,k = A k (1 j μ), (1 k μ). (6) Thus, taking μ = 1,2,..., 1,wecanassertthatc () jk are linear cobinations of operators A k and their products. This proves the required result. But T has the for B,1, T 2 has the for B,2,andsoforth. Take B,μ = T μ.then,d,μ+1 = T μ+1.denotebyc (μ) jk the entries of T μ. Then according to (6), c (μ+1) jk = c (μ) j1 A k +c (μ) j,k+1 (k <), For linear operators A and A, wewillsaythatsv A ( A) := sup μ σ( A) inf s σ(a) μ s is the spectral variation of A with respect to A. Lea 2. One has zsv (j) P ( P )=sv T j( T j ) (j=1,2,...,). Proof. As it is well known, cf. [1, the spectra of T and P ( ) coincide. This iplies the required result.

3 Coplex Analysis 3 2. The Main Results The nor H in H is defined by the following way: let H= k=1 H k,h k H(k=1,...,). Then, f 2 H = k=1 f k 2 (f = (f k H k ) k=1 H ). (8) Put q = T T H and assue that, for an integer p 1, A k SN 2p (k=1,...,). (9) Theore 3. Let condition (9) hold. Then, zsv () P ( P ) y,p, y,p istheuniquepositiverootofequation p 1(2N 2p (T )) s q z s+1 exp [ (2N 2p(T 2p )) 2z 2p =1. (10) ThisresultisduetoTheore8.5.4 of [14andLea2. Due to Corollary of [14, we have zsv () P ( P ) Δ p,, (11) Reark 5. Put I T =(T (T ) )/2i. Then according to Theore of [14, in Theores 3 and 4, onecanreplace 2N 2p (T ) by β p N 2p (I T ), β p := 2 ( Proof of Theore 4 2p ). (17) exp (2/3) ln 2 Let A and A be bounded linear operators in H, E= A A and Z= AE EA. We begin with the following result. Lea 6. Let a λ C be regular for both A and A.Then,the following equality holds: R λ ( A) R λ (A) =R λ ( A) ZR 2 λ (A) ER2 λ (A). (18) Proof. We have R λ ( A) ( AE EA) R 2 λ (A) ER2 λ (A) =(R λ ( A) ( AE EA) E) R 2 λ (A) peq if 2N 2p (T ) epq, Δ p, := 2N 2p (T )[ln ( 2N 2p (T 1/2p ) (12) ) q p if 2N 2p (T )>epq. Now consider perturbations of pencils with alost couting coefficients. To this end, put Ξ = T T and ζ (T, Ξ ) := T Ξ Ξ T + Ξ2. (13) =R λ ( A) ( AE EA ( A Iλ)E)R 2 λ (A) = R λ ( A) ( Eλ + EA) R 2 λ (A) = R λ ( A) ER λ (A) = R λ ( A) ( A Iλ (A Iλ))R λ (A) = (I R λ ( A) (A Iλ))R λ (A) =R λ ( A) R λ (A), (19) Theore 4. Let condition (9) hold. Then, zsv () P ( P ) x,p, x,p istheuniquepositiverootofequation p 1(2N 2p (T )) s ζ (T, Ξ ) z s+1 exp [ (2N 2p (T )) 2p 2z 2p [ =1. (14) This theore is proved in the next section. Replacing in (9) q by ζ := ζ (T, Ξ ),weobtain zsv () P ( P ) δ p,, (15) peζ as claied. Denote η(a, E, λ) = sup 0 t 1 t (AE EA + te 2 )R 2 λ (A). Corollary 7. Let a λ C be regular for A and Then, λ is regular also for A. η (A, E, λ) <1. (20) Indeed, put A t = A+tE (t [0, 1). Since the regular sets of operators are open, λ is regular for A t,providedt is sall enough. By Lea 6, R λ (A t ) R λ (A) =R λ (A t ) (t (A+tE) E tea) R 2 λ (A) ter 2 λ (A). (21) if 2N 2p (T ) epζ, δ p, := 2N 2p (T )[ln ( 2N 2p (T ) ) ζ p if 2N 2p (T )>epζ. 1/2p (16) It is clear that Theore 4 is sharper than Theore 3,provided ζ (T, Ξ )<q. Hence, R λ (A t ) R λ (A) ter 2 λ (A) R λ (A t ) [t (EA AE) +t2 E 2 R 2 λ (A) R λ (A t ) η (A, E, λ). (22)

4 4 Coplex Analysis Thus, R λ (A t ) R λ (A) ter 2 λ (A). (23) 1 η(a, E, λ) 4. Quadratic Pencils In this section, =2.So,P 2 (z) = z 2 +A 1 z+a 2, P 2 (z) = z 2 + A 1 z+ A 2, Taking t=1, we obtain the required result. Put ζ(a, E) := AE EA + E 2.Itisclearthatη(A, E, λ) ζ 2 (A, E) R 2 λ (A). Now Corollary 7 iplies the following. Corollary 8. If λ is regular for A and ζ(a, E) R λ (A) < 1, then, λ is regular also for A. T=T(P 2 )=( A 1 A 2 I 0 ), T 2 =( A2 1 A 2 A 1 A 2 A 1 A 2 ), T 2 =( A 2 1 A 2 A 1 A 2 A 1 A 2 ). (29) Furtherore, assue that A SN 2p (p=1,2,...). (24) Then due to Theore of [14, we have the estiate p 1 R λ (A) (2N 2p (A)) s ρ s+1 (A, λ) exp [ (2N 2p 2p (A)) 2ρ 2p (A, λ), [ (25) ρ(a, λ) = inf s σ(a) s λ the distance between λ and the spectru ofa.now Corollary8 iplies the following. Corollary 9. If condition (24) holds, λ is regular for A and p 1(2N 2p (A)) s ζ (A, E) ρ s+1 (A, λ) exp [ (2N 2p 2p (A)) 2ρ 2p (A, λ) <1, [ (26) then, λ is regular also for A. Lea 10. Let condition (24) hold. Then, sv A ( A) z p, z p istheuniquepositiverootofequation p 1(2N 2p (A)) s ζ (A, E) z s+1 exp [ (2N 2p 2p(A)) 2z 2p =1. (27) Proof. For any μ σ(a),duetothepreviouscorollary,we have p 1(2N 2p (A)) s ζ (A, E) ρ s+1 (A, μ) exp [1 2 + (2N 2p 2p(A)) 2ρ 2p (A, μ) 1. (28) Hence, it follows that ρ(a, μ) z p. But sv A ( A) = sup μ σ( A) ρ(a, μ).wethusarriveattherequiredresult. 2. The assertion of Theore 4 follows fro Leas 10 and Now we can directly apply Theores 3 and 4. To derive bounds for the spectru of P 2,takeanoperator B couting with A 1. For exaple, B=cA 2 1 with a constant c.ifitisdesirabletochoosebinsuch a way that the nor of A 2 Bis sall enough. Put Q(z) = z 2 +A 1 z+band S=( A 1 B I 0 ). Then, S2 =( A2 1 B A 1B ). (30) A 1 B Since A 2 and B coute, one can enuerate their eigenvalues in such a way that the eigenvalues of Q(z) for a fixed z are z 2 +λ k (A 1 )z + λ k (B).So, Σ(Q) = z k1 (Q), z k2 (Q)} k=1, z k1 (Q), z k2 (Q) are the roots of the polynoial z 2 +λ k (A 1 )z + λ k (B): We have z k1 (Q) = λ k (A 1 ) 2 z k2 (Q) = λ k (A 1 ) 2 + λ2 k (A 1) 4 λ2 k (A 1) 4 λ k (B), λ k (B). (31) T 2 (P 2 ) S 2 =( C A 1C ), (32) 0 C C=A 2 B.So T 2 (P 2 ) S 2 H 2 q S := C (1 + A 1 ). Now inequality (11) iplies the following. Corollary 11. Let A 1,B SN 2p, (33) and let B coute with A 1.Thenforanynonzeros Σ(P 2 ), there is a z kl (Q) (l = 1,2; k = 1,2,...), such that s 2 z 2 k,l δ(s), peq S if 2N 2p (S 2 ) epq S, δ (S) := 2N 2p (S 2 )[ln ( 2N 2p (S 2 ) ) q S p if 2N 2p (S 2 )>epq S. 1/2p (34)

5 Coplex Analysis 5 Let r s (P 2 ) be the spectral radius of P 2 : r s (P 2 ) := sup s Σ(P2 ) s. Fro the previous corollary, it follows that r 2 s (P 2) r 2 s (Q) + δ(s). Besides, r s (Q) = Corollary 12. Under condition (33),let Then, r s (P 2 )<1. r 2 s sup z kl (Q). (35) l=1,2; k=1,2,... (Q) +δ(s) <1. (36) 5. Difference and Differential Equations In the present section, we briefly discuss applications of our results to difference and differential equations. Consider the difference equation k=0 A k V (j + k) = 0 (A 0 =I; j=,+1,...), (37) with bounded operator coefficients A k. This equation is said to be asyptotically stable, if any of its solution tends to zero as t. It is not hard to check that (37)isasyptotically stable, provided r s (P )<1; (38) cf. [13. Now one can use the perturbation results due to Theores 3 and 4. For exaple, let =2.So, V (j+2) +A 1 V (j+1) +A 2 V (j) =0 (j = 2, 3,...). (39) Take an operator B couting with A 1 as in the previous section. Recall that it is desirable to choose B such that the nor of A 2 Bis sall enough. Now Corollary 12 iplies the following. Corollary 13. Under conditions (33) and (36), (39) is asyptotically stable. Furtherore, let us consider in H the differential equation u xx (x) +A 1 u x (x) +A 2 u (x) =f(x) (0<x<1), (40) f (x) = f k e 2πikx L 2 ([0, 1,H) (f k H). (41) k= Nuerous integrodifferential equations can be written in the for of (40). Ipose the periodic conditions u (0) =u(1), u (0) =u (1). (42) We seek a solution of proble (40), (42)inthefor u (x) = u k e 2πikx, (43) k= u k (k=0,±1,...)shouldbefound.substitutingthis expression into (40), we obtain ( 4π 2 k 2 I+2πikA 1 +A 2 )u k =f k. (44) This equation has a solution provided the spectru of P 2 (z) does not contain the nubers 2πik. Now one can apply Corollary 11. References [1 L. Rodan, An Introduction to Operator Polynoials,vol.38of Operator Theory: Advances and Applications,Birkhäuser, Basel, Switzerland, [2 E. Bairaov, Ö. Çakar,andA.M.Krall, Spectralproperties, including spectral singularities, of a quadratic pencil of Schrödinger operators on the whole real axis, Quaestiones Matheaticae,vol.26,no.1,pp.15 30,2003. [3 P. A. Cojuhari, Estiates of the discrete spectru of a linear operator pencil, Matheatical Analysis and Applications, vol. 326, no. 2, pp , [4 R. F. Efendiev, Spectral analysis for one class of secondorder indefinite non-self-adjoint differential operator pencil, Applicable Analysis,vol.90,no.12,pp ,2011. [5 M.I.Gil, OnboundsforspectraofoperatorpencilsinaHilbert space, Acta Matheatica Sinica (English Series), vol. 19, no. 2, pp , [6 M. I. Gil, Bounds for the spectru of analytic quasinoral operator pencils, Counications in Conteporary Matheatics,vol.5,no.1,pp ,2003. [7 M. I. Gil, Sus of characteristic values of copact polynoial operator pencils, Matheatical Analysis and Applications,vol.338,no.2,pp ,2008. [8 M. Hasanov, An approxiation ethod in the variational theory of the spectru of operator pencils, Acta Applicandae Matheaticae,vol.71,no.2,pp ,2002. [9 I. V. Kurbatova, A Banach algebra associated with a linear operator pencil, Mateaticheskie Zaetki, vol. 86, no. 3, pp , 2009 (Russian), translation in Matheatical Notes, vol. 86 (2009), no. 3-4, [10 I. V. Kurbatova, A functional calculus generated by a quadratic operator pencil, Matheatical Sciences, vol.182,no. 5,pp ,2012. [11 M. D. Manafov and A. Kablan, On a quadratic pencil of differential operators with periodic generalized potential, International Pure and Applied Matheatics,vol.50,no.4, pp , [12 Y. Yakubov, Fold copleteness of a syste of root vectors of a syste of unbounded polynoial operator pencils in Banach spaces. I. Abstract theory, Journal de Mathéatiques Pures et Appliquées, vol. 92, no. 3, pp , [13 M. I. Gil, Difference Equations in Nored Spaces: Stability and Oscillations, vol.206ofnorth-holland Matheatics Studies, Elsevier Science B.V., Asterda, The Netherlands, [14 M. I. Gil, Operator Functions and Localization of Spectra, vol of Lecture Notes in Matheatics, Springer, Berlin, Gerany, 2003.

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