Perturbations of Polynomials with Operator Coefficients

Hindawi Publishing Corporation Journal of Complex Analysis Volume 2013, Article ID 801382, 5 pages http://dx.doi.org/10.1155/2013/801382 Research Article Perturbations of Polynomials with Operator Coefficients Michael Gil’ Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, 84105 Beer-Sheva, Israel Correspondence should be addressed to Michael Gil’; [email protected] Received 18 December 2012; Accepted 20 February 2013 Academic Editor: Janne Heittokangas Copyright © 2013 Michael Gil’. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider polynomials whose coefficients are operators belonging to the Schatten-von Neumann ideals of compact 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 Preliminaries Numerous mathematical and physical problems lead to polynomial operator pencils (polynomials with operator coefficients); cf. [1] and references therein. Recently, the spectral theory of operator pencils attracts the attention of many mathematicians. 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 the discrete spectrum of linear pencils. The paper [4] deals with the spectral analysis of a class of second-order indefinite nonself-adjoint differential operator pencils. In that paper, a method for solving the inverse spectral problem for the Schrődinger operator with complex 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 polynomial pencils whose coefficients are compact operators. Besides, inequalities for the sums of absolute values and real and imaginary 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 completeness of a system of root vectors of a system of unbounded polynomial 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 many applications, for example, in numerical mathematics and stability analysis, bounds for the spectra of perturbed pencils are very important; cf. [13]. In the present paper, we derive such bounds in the case of polynomials whose coefficients are operators belonging to the Schattenvon Neumann ideals of compact operators in a Hilbert space. Introduce the notations. Let 𝐻 be a separable complex Hilbert space with a scalar product (⋅, ⋅) and the norm ‖ ⋅ ‖ = √(⋅, ⋅). By 𝐼 the identity operator in 𝐻 is denoted. For a linear operator 𝐴 in 𝐻, 𝐴−1 is the inverse operator, 𝜎(𝐴) is the spectrum, 𝜆 𝑘 (𝐴) (𝑘 = 1, 2, . . .) are the eigenvalues with their multiplicities, 𝐴∗ is the adjoint operator, ‖𝐴‖ is the operator norm, and 𝑅𝜆 (𝐴) = (𝐴 − 𝜆𝐼)−1 is the resolvent. SN] (1 ≤ ] < ∞) is the Schatten-von Neumann ideal of compact operators 𝐾 in 𝐻 having the finite norm 𝑁] (𝐾) = [Trace(𝐾𝐾∗ )]/2 ]1/] . ̃𝑘 (𝑘 = 1, . . . , 𝑚 < ∞) be linear bounded Let 𝐴 𝑘 and 𝐴 operators in 𝐻. Consider the pencils 𝑚 𝑃𝑚 (𝜆) = ∑ 𝐴 𝑘 𝜆𝑚−𝑘 , 𝑘=0 𝑚 ̃𝑘 𝜆𝑚−𝑘 ̃ 𝑚 (𝜆) = ∑ 𝐴 𝑃 𝑘=0 ̃0 = 𝐴 0 = 𝐼) . (𝐴 (1) 2 Journal of Complex Analysis A point 𝜆 ∈ C is called a regular value of 𝑃𝑚 , if 𝑃𝑚 (𝜆) is boundedly invertible. The complement of all regular points of 𝑃𝑚 to the closed complex plane is called the spectrum of 𝑃𝑚 and is denoted by Σ(𝑃𝑚 ). Our main problem is as follows: if ̃𝑘 (𝑘 = 1, . . . , 𝑚) are close, how close are the spectra 𝐴 𝑘 and 𝐴 ̃ 𝑚 (𝜆)? of 𝑃𝑚 (𝜆) and 𝑃 For an integer 𝑗 ≥ 1, we will say that (𝑗) ̃ 𝑚 ) := sup 𝑧𝑠V𝑃𝑚 (𝑃 ̃𝑚) 𝜇∈Σ(𝑃 󵄨 󵄨 inf 󵄨󵄨󵄨󵄨𝜇𝑗 − 𝑠𝑗 󵄨󵄨󵄨󵄨 𝑠∈Σ(𝑃𝑚 ) (2) ̃ 𝑚 with respect to 𝑃𝑚 . Let 𝑇 = is the 𝑗-spectral variation of 𝑃 ̃ = 𝑇(𝑃 ̃ 𝑚 ) be the operator 𝑚 × 𝑚-matrices 𝑇(𝑃𝑚 ) and 𝑇 defined on the the orthogonal sum 𝐻𝑚 of 𝑚 exemplars of 𝐻 by −𝐴 1 −𝐴 2 𝐼 0 𝐼 𝑇=( 0 ⋅ ⋅ 0 0 ⋅ ⋅ ⋅ −𝐴 𝑚−1 −𝐴 𝑚 ⋅⋅⋅ 0 0 ⋅⋅⋅ 0 0 ), ⋅⋅⋅ ⋅ ⋅ ⋅⋅⋅ 𝐼 0 𝐷𝑚,𝜇+1 𝑑12 𝑑11 𝑑21 𝑑22 ⋅ ( ⋅ ( 𝑑𝜇1 𝑑𝜇2 ( =( (𝑑𝜇+1,1 𝑑𝜇+1,2 ( 𝐼 0 ( 0 𝐼 ⋅ ⋅ 0 0 ( ̃2 ̃ 1 −𝐴 −𝐴 𝐼 0 ̃ =( 0 𝑇 𝐼 ⋅ ⋅ 0 0 Lemma 1. Let all the operators 𝐴 𝑘 (𝑘 = 1, . . . , 𝑚) belong to some ideal 𝐽 of compact operators. Then, all the entries of the operator matrix 𝑇𝑚 also belong to 𝐽. Proof. For an integer 𝜇 < 𝑚, consider the operator 𝑚 × 𝑚 matrix 𝐵𝑚,𝜇 𝑏11 𝑏12 𝑏21 𝑏22 ⋅ (⋅ (𝑏𝜇1 𝑏𝜇,2 =( (𝐼 0 ( 0 𝐼 ⋅ ⋅ (0 0 ⋅ ⋅ ⋅ 𝑑1,𝑚−𝜇−1 ⋅ ⋅ ⋅ 𝑑2,𝑚−𝜇−1 ⋅⋅⋅ ⋅ ⋅ ⋅ ⋅ 𝑑𝜇,𝑚−𝜇−1 ⋅ ⋅ ⋅ 𝑑𝜇+1,𝑚−𝜇−1 ⋅⋅⋅ 0 ⋅⋅⋅ 0 ⋅⋅⋅ ⋅ ⋅⋅⋅ 𝐼 (𝜇+1) 𝑑𝑗𝑚 = −𝑏𝑗1 𝐴 𝑚 (1 ≤ 𝑗 ≤ 𝜇) , ⋅ ⋅ ⋅ 𝑏1,𝑚−𝜇 ⋅ ⋅ ⋅ 𝑏2,𝑚−𝜇 ⋅⋅⋅ ⋅ ⋅ ⋅ ⋅ 𝑏𝜇,𝑚−𝜇 ⋅⋅⋅ 0 ⋅⋅⋅ 0 ⋅⋅⋅ ⋅ ⋅⋅⋅ 𝐼 ⋅ ⋅ ⋅ 𝑏1,𝑚−1 𝑏1,𝑚 ⋅ ⋅ ⋅ 𝑏2,𝑚−1 𝑏2,𝑚 ⋅⋅⋅ ⋅ ⋅ ) ⋅ ⋅ ⋅ 𝑏𝜇,𝑚−1 𝑏𝜇,𝑚 ) ) ⋅⋅⋅ 0 0 ) ) ⋅⋅⋅ 0 0 ⋅⋅⋅ ⋅ ⋅ ⋅⋅⋅ 0 0 ) ⋅ ⋅ ⋅ 𝑑1,𝑚−1 𝑑1,𝑚 ⋅ ⋅ ⋅ 𝑑2,𝑚−1 𝑑2,𝑚 ⋅⋅⋅ ⋅ ⋅ ) ⋅ ⋅ ⋅ 𝑑𝑚𝑢,𝑚−1 𝑑𝜇,𝑚 ) ) ⋅ ⋅ ⋅ 𝑑𝜇+1,𝑚−1 𝑑𝜇+1,𝑚 ) ), ⋅⋅⋅ 0 0 ) ) ⋅⋅⋅ 0 0 ⋅⋅⋅ ⋅ ⋅ ⋅⋅⋅ 0 0 ) 𝑐𝑗𝜇 (1 ≤ 𝑗 ≤ 𝜇, 𝑘 < 𝑚) , (3) (4) with some operators 𝑏𝑗𝑘 . Direct calculations show that the operator matrix 𝐷𝑚,𝜇+1 := 𝐵𝑚,𝜇 𝑇 has the form where 𝑑𝑗𝑘 = −𝑏𝑗1 𝐴 𝑘 + 𝑏𝑗,𝑘+1 ̃𝑚−1 −𝐴 ̃𝑚 ⋅ ⋅ ⋅ −𝐴 ⋅⋅⋅ 0 0 ⋅⋅⋅ 0 0 ). ⋅⋅⋅ ⋅ ⋅ ⋅⋅⋅ 𝐼 0 (𝜇) = −𝑐𝑗1 𝐴 𝑚 (𝜇+1) (5) (1 ≤ 𝑗 ≤ 𝜇) , 𝑐𝜇+1,𝑘 = −𝐴 𝑘 . (7) (6) (𝑚) Thus, taking 𝜇 = 1, 2, . . . , 𝑚 − 1, we can assert that 𝑐𝑗𝑘 are linear combinations of operators 𝐴 𝑘 and their products. This proves the required result. But 𝑇 has the form 𝐵𝑚,1 , 𝑇2 has the form 𝐵𝑚,2 , and so forth. (𝜇) Take 𝐵𝑚,𝜇 = 𝑇𝜇 . Then, 𝐷𝑚,𝜇+1 = 𝑇𝜇+1 . Denote by 𝑐𝑗𝑘 the entries of 𝑇𝜇 . Then according to (6), ̃ we will say that 𝑠V𝐴 (𝐴) ̃ := For linear operators 𝐴 and 𝐴, ̃ sup𝜇∈𝜎(𝐴) inf |𝜇 − 𝑠| is the spectral variation of 𝐴 with ̃ 𝑠∈𝜎(𝐴) respect to 𝐴. 𝑑𝜇+1,𝑘 = −𝐴 𝑘 (𝜇+1) 𝑐𝑗𝑘 (𝜇) (1 ≤ 𝑘 ≤ 𝜇) . (𝜇) = −𝑐𝑗1 𝐴 𝑘 + 𝑐𝑗,𝑘+1 (𝑘 < 𝑚) , (𝑗) 𝑗 ̃ ) (𝑗 = 1, 2, . . . , 𝑚). ̃ 𝑚 ) = 𝑠V𝑇𝑗 (𝑇 Lemma 2. One has 𝑧𝑠V𝑃𝑚 (𝑃 Proof. As it is well known, cf. [1], the spectra of 𝑇 and 𝑃𝑚 (⋅) coincide. This implies the required result. Journal of Complex Analysis 3 2. The Main Results The norm ‖ ⋅ ‖𝐻𝑚 in 𝐻𝑚 is defined by the following way: let 𝐻 = ∑𝑚 𝑘=1 ⊕ 𝐻𝑘 , where 𝐻𝑘 ≡ 𝐻 (𝑘 = 1, . . . , 𝑚). Then, ‖𝑓‖2𝐻𝑚 𝑚 = ∑ ‖𝑓𝑘 ‖ 2 𝑘=1 (𝑓 = (𝑓𝑘 ∈ 𝐻𝑘 )𝑚 𝑘=1 𝑚 ∈ 𝐻 ). 𝐴 𝑘 ∈ SN2𝑝 (𝑘 = 1, . . . , 𝑚) . (9) ̃ 𝑚 ) ≤ 𝑦𝑚,𝑝 , (𝑃 Theorem 3. Let condition (9) hold. Then, 𝑧𝑠V𝑃(𝑚) 𝑚 where 𝑦𝑚,𝑝 is the unique positive root of equation 𝑝−1 𝑞𝑚 ∑ 𝑠=0 𝑚 (2𝑁2𝑝 (𝑇 )) 𝑧𝑠+1 𝑠 𝑚 2𝑝 1 (2𝑁2𝑝 (𝑇 )) exp [ + ] = 1. 2 2𝑧2𝑝 (10) This result is due to Theorem 8.5.4 of [14] and Lemma 2. Due to Corollary 8.5.5 of [14], we have ̃ 𝑚 ) ≤ Δ 𝑝,𝑚 , (𝑃 𝑧𝑠V𝑃(𝑚) 𝑚 where (11) 𝑝𝑒𝑞𝑚 { { { { if 2𝑁2𝑝 (𝑇𝑚 ) ≤ 𝑒𝑝𝑞𝑚 , { { { −1/2𝑝 Δ 𝑝,𝑚 := { (12) 2𝑁2𝑝 (𝑇𝑚 ) 𝑚 { )] 2𝑁 (𝑇 ) [ln ( { 2𝑝 { { 𝑞𝑚 𝑝 { { 𝑚 if 2𝑁 (𝑇 ) > 𝑒𝑝𝑞 2𝑝 𝑚. { Now consider perturbations of pencils with almost commuting coefficients. ̃ 𝑚 − 𝑇𝑚 and To this end, put Ξ𝑚 = 𝑇 ̂𝜁 (𝑇, Ξ ) := √󵄩󵄩󵄩𝑇𝑚 Ξ − Ξ 𝑇𝑚 󵄩󵄩󵄩 + 󵄩󵄩󵄩Ξ2 󵄩󵄩󵄩. 𝑚 𝑚 𝑚 𝑚 󵄩 󵄩 󵄩 𝑚󵄩 (13) ̃ 𝑚 ) ≤ 𝑥𝑚,𝑝 , Theorem 4. Let condition (9) hold. Then, 𝑧𝑠V𝑃(𝑚) (𝑃 𝑚 where 𝑥𝑚,𝑝 is the unique positive root of equation 𝑝−1 (2𝑁 (𝑇𝑚 )) 2𝑝 𝑧𝑠+1 𝑠=0 ̂𝜁 (𝑇, Ξ ) ∑ 𝑚 𝑚 = 1. 𝑠 2𝑝 𝑚 1 (2𝑁2𝑝 (𝑇 )) ] exp [ + 2 2𝑧2𝑝 ] [ (14) This theorem is proved in the next section. Replacing in (9) 𝑞𝑚 by 𝜁𝑚 := ̂𝜁𝑚 (𝑇, Ξ𝑚 ), we obtain where 𝛿𝑝,𝑚 ̃ 𝑚 ) ≤ 𝛿𝑝,𝑚 , (𝑃 𝑧𝑠V𝑃(𝑚) 𝑚 𝑝𝑒𝜁𝑚 { { { { 𝑚 { { { { if 2𝑁2𝑝 (𝑇 ) ≤ 𝑒𝑝𝜁𝑚 , −1/2𝑝 := { 2𝑁2𝑝 (𝑇𝑚 ) { 𝑚 { { 2𝑁2𝑝 (𝑇 ) [ln ( )] { { 𝜁𝑚 𝑝 { { 𝑚 { if 2𝑁2𝑝 (𝑇 ) > 𝑒𝑝𝜁𝑚 . 𝛽𝑝 := 2 (1 + (8) ̃ 𝑚 ‖𝐻𝑚 and assume that, for an integer 𝑝 ≥ 1, Put 𝑞𝑚 = ‖𝑇 − 𝑇 𝑚 Remark 5. Put Im 𝑇𝑚 = (𝑇𝑚 − (𝑇∗ )𝑚 )/2𝑖. Then according to Theorem 7.9.1 of [14], in Theorems 3 and 4, one can replace 2𝑁2𝑝 (𝑇𝑚 ) by 𝛽𝑝 𝑁2𝑝 (Im 𝑇𝑚 ), where (15) 3. Proof of Theorem 4 2𝑝 ). exp (2/3) ln 2 ̃ be bounded linear operators in 𝐻, 𝐸 = 𝐴 ̃−𝐴 Let 𝐴 and 𝐴 ̃ − 𝐸𝐴. We begin with the following result. and 𝑍 = 𝐴𝐸 ̃ Then, the Lemma 6. Let a 𝜆 ∈ C be regular for both 𝐴 and 𝐴. following equality holds: ̃ − 𝑅𝜆 (𝐴) = 𝑅𝜆 (𝐴) ̃ 𝑍𝑅2 (𝐴) − 𝐸𝑅2 (𝐴) . 𝑅𝜆 (𝐴) 𝜆 𝜆 It is clear that Theorem 4 is sharper than Theorem 3, provided ̂𝜁 (𝑇, Ξ ) < 𝑞 . 𝑚 𝑚 𝑚 (18) Proof. We have ̃ (𝐴𝐸 ̃ − 𝐸𝐴) 𝑅2 (𝐴) − 𝐸𝑅2 (𝐴) 𝑅𝜆 (𝐴) 𝜆 𝜆 ̃ (𝐴𝐸 ̃ − 𝐸𝐴) − 𝐸) 𝑅2 (𝐴) = (𝑅𝜆 (𝐴) 𝜆 ̃ (𝐴𝐸 ̃ − 𝐸𝐴 − (𝐴 ̃ − 𝐼𝜆) 𝐸) 𝑅2 (𝐴) = 𝑅𝜆 (𝐴) 𝜆 ̃ (−𝐸𝜆 + 𝐸𝐴) 𝑅2 (𝐴) = −𝑅𝜆 (𝐴) 𝜆 ̃ 𝐸𝑅𝜆 (𝐴) = −𝑅𝜆 (𝐴) (19) ̃ (𝐴 ̃ − 𝐼𝜆 − (𝐴 − 𝐼𝜆)) 𝑅𝜆 (𝐴) = −𝑅𝜆 (𝐴) ̃ (𝐴 − 𝐼𝜆)) 𝑅𝜆 (𝐴) = − (𝐼 − 𝑅𝜆 (𝐴) ̃ − 𝑅𝜆 (𝐴) , = 𝑅𝜆 (𝐴) as claimed. Denote 𝜂(𝐴, 𝐸, 𝜆) = sup0≤𝑡≤1 𝑡‖(𝐴𝐸 − 𝐸𝐴 + 𝑡𝐸2 )𝑅𝜆2 (𝐴)‖. Corollary 7. Let a 𝜆 ∈ C be regular for 𝐴 and 𝜂 (𝐴, 𝐸, 𝜆) < 1. (20) ̃ Then, 𝜆 is regular also for 𝐴. Indeed, put 𝐴 𝑡 = 𝐴 + 𝑡𝐸 (𝑡 ∈ [0, 1]). Since the regular sets of operators are open, 𝜆 is regular for 𝐴 𝑡 , provided 𝑡 is small enough. By Lemma 6, 𝑅𝜆 (𝐴 𝑡 ) − 𝑅𝜆 (𝐴) = 𝑅𝜆 (𝐴 𝑡 ) (𝑡 (𝐴 + 𝑡𝐸) 𝐸 − 𝑡𝐸𝐴) 𝑅𝜆2 (𝐴) − 𝑡𝐸𝑅𝜆2 (𝐴) . (16) (17) Hence, 󵄩 󵄩󵄩 󵄩 󵄩 2 󵄩󵄩𝑅𝜆 (𝐴 𝑡 )󵄩󵄩󵄩 − 󵄩󵄩󵄩󵄩𝑅𝜆 (𝐴) − 𝑡𝐸𝑅𝜆 (𝐴)󵄩󵄩󵄩󵄩 󵄩 󵄩 󵄩󵄩 ≤ 󵄩󵄩󵄩𝑅𝜆 (𝐴 𝑡 )󵄩󵄩󵄩 󵄩󵄩󵄩󵄩[𝑡 (𝐸𝐴 − 𝐴𝐸) + 𝑡2 𝐸2 ] 𝑅𝜆2 (𝐴)󵄩󵄩󵄩󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩𝑅𝜆 (𝐴 𝑡 )󵄩󵄩󵄩 𝜂 (𝐴, 𝐸, 𝜆) . (21) (22) 4 Journal of Complex Analysis 4. Quadratic Pencils Thus, 󵄩󵄩 󵄩󵄩 2 󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩𝑅𝜆 (𝐴) − 𝑡𝐸𝑅𝜆 (𝐴)󵄩󵄩󵄩 . 󵄩󵄩𝑅𝜆 (𝐴 𝑡 )󵄩󵄩 ≤ 1 − 𝜂 (𝐴, 𝐸, 𝜆) (23) Taking 𝑡 = 1, we obtain the required result. Put 𝜁(𝐴, 𝐸) := √‖𝐴𝐸 − 𝐸𝐴‖ + ‖𝐸2 ‖. It is clear that 𝜂(𝐴, 𝐸, 𝜆) ≤ 𝜁2 (𝐴, 𝐸)‖𝑅𝜆2 (𝐴)‖. Now Corollary 7 implies the following. Corollary 8. If 𝜆 is regular for 𝐴 and 𝜁(𝐴, 𝐸)‖𝑅𝜆 (𝐴)‖ < 1, ̃ then, 𝜆 is regular also for 𝐴. Furthermore, assume that 𝐴 ∈ SN2𝑝 (𝑝 = 1, 2, . . .) . (24) Then due to Theorem 7.9.1 of [14], we have the estimate 𝑠 2𝑝 𝑝−1 (2𝑁 1 (2𝑁2𝑝 (𝐴)) ] 2𝑝 (𝐴)) 󵄩 󵄩󵄩 , exp [ + 󵄩󵄩𝑅𝜆 (𝐴)󵄩󵄩󵄩 ≤ ∑ 𝑠+1 2 2𝜌2𝑝 (𝐴, 𝜆) (𝐴, 𝜆) 𝑠=0 𝜌 [ ] (25) where 𝜌(𝐴, 𝜆) = inf 𝑠∈𝜎(𝐴) |𝑠 − 𝜆|—the distance between 𝜆 and the spectrum of 𝐴. Now Corollary 8 implies the following. Corollary 9. If condition (24) holds, 𝜆 is regular for 𝐴 and 𝑝−1 (2𝑁 2𝑝 (𝐴)) 𝜁 (𝐴, 𝐸) ∑ 𝑠+1 (𝐴, 𝜆) 𝑠=0 𝜌 𝑠 ̃ then, 𝜆 is regular also for 𝐴. 2𝑝 1 (2𝑁2𝑝 (𝐴)) ] < 1, exp [ + 2 2𝜌2𝑝 (𝐴, 𝜆) [ ] (26) 2𝑝 1 (2𝑁2𝑝 (𝐴)) exp [ + ] = 1. 2 2𝑧2𝑝 (27) Proof. For any 𝜇 ∈ 𝜎(𝐴), due to the previous corollary, we have 𝑠 𝑝−1 (2𝑁 2𝑝 (𝐴)) 𝜁 (𝐴, 𝐸) ∑ 𝑠+1 (𝐴, 𝜇) 𝑠=0 𝜌 2𝑝 1 (2𝑁2𝑝 (𝐴)) exp [ + ] ≥ 1. 2 2𝜌2𝑝 (𝐴, 𝜇) (28) ̃ Hence, it follows that 𝜌(𝐴, 𝜇) ≤ 𝑧𝑝 . But 𝑠V𝐴(𝐴) 𝜌(𝐴, 𝜇). We thus arrive at the required result. sup𝜇∈𝜎(𝐴) ̃ = The assertion of Theorem 4 follows from Lemmas 10 and 2. −𝐴 −𝐴 2 ), 𝑇 = 𝑇 (𝑃2 ) = ( 1 𝐼 0 𝐴2 − 𝐴 2 𝐴 1 𝐴 2 𝑇2 = ( 1 ), −𝐴 1 −𝐴 2 (29) ̃2 − 𝐴 ̃1 𝐴 ̃2 ̃2 𝐴 ̃ 2 = (𝐴 1 𝑇 ). ̃1 ̃2 −𝐴 −𝐴 Now we can directly apply Theorems 3 and 4. To derive bounds for the spectrum of 𝑃2 , take an operator 𝐵 commuting with 𝐴 1 . For example, 𝐵 = 𝑐𝐴21 with a constant 𝑐. If it is desirable to choose 𝐵 in such a way that the norm of 𝐴 2 − 𝐵 is small enough. Put 𝑄(𝑧) = 𝑧2 + 𝐴 1 𝑧 + 𝐵 and −𝐴 −𝐵 𝑆=( 1 ). 𝐼 0 𝐴2 − 𝐵 𝐴 1 𝐵 ) . (30) Then, 𝑆2 = ( 1 −𝐴 1 −𝐵 Since 𝐴 2 and 𝐵 commute, one can enumerate their eigenvalues in such a way that the eigenvalues of 𝑄(𝑧) for a fixed 𝑧 are 𝑧2 + 𝜆 𝑘 (𝐴 1 )𝑧 + 𝜆 𝑘 (𝐵). So, Σ(𝑄) = {𝑧𝑘1 (𝑄), 𝑧𝑘2 (𝑄)}∞ 𝑘=1 , where 𝑧𝑘1 (𝑄), 𝑧𝑘2 (𝑄) are the roots of the polynomial 𝑧2 + 𝜆 𝑘 (𝐴 1 )𝑧 + 𝜆 𝑘 (𝐵): 𝑧𝑘1 (𝑄) = − We have ̃ ≤ 𝑧𝑝 , where Lemma 10. Let condition (24) hold. Then, 𝑠V𝐴(𝐴) 𝑧𝑝 is the unique positive root of equation 𝑠 𝑝−1 (2𝑁 2𝑝 (𝐴)) 𝜁 (𝐴, 𝐸) ∑ 𝑧𝑠+1 𝑠=0 ̃ 2 (𝑧) = In this section, 𝑚 = 2. So, 𝑃2 (𝑧) = 𝑧2 + 𝐴 1 𝑧 + 𝐴 2 , 𝑃 2 ̃ ̃ 𝑧 + 𝐴1 𝑧 + 𝐴2 , 𝑧𝑘2 (𝑄) = − 𝜆 𝑘 (𝐴 1 ) √ 𝜆2𝑘 (𝐴 1 ) + − 𝜆 𝑘 (𝐵), 2 4 𝜆 𝑘 (𝐴 1 ) − 2 2 √ 𝜆𝑘 (31) (𝐴 1 ) − 𝜆 𝑘 (𝐵). 4 −𝐶 𝐴 1 𝐶 𝑇2 (𝑃2 ) − 𝑆2 = ( ), 0 −𝐶 (32) where 𝐶 = 𝐴 2 − 𝐵. So ‖𝑇2 (𝑃2 ) − 𝑆2 ‖𝐻2 ≤ 𝑞𝑆 := ‖𝐶‖(1 + ‖𝐴 1 ‖). Now inequality (11) implies the following. Corollary 11. Let 𝐴 1 , 𝐵 ∈ 𝑆𝑁2𝑝 , (33) and let 𝐵 commute with 𝐴 1 . Then for any nonzero 𝑠 ∈ Σ(𝑃2 ), 2 there is a 𝑧𝑘𝑙 (𝑄) (𝑙 = 1, 2; 𝑘 = 1, 2, . . .), such that |𝑠2 − 𝑧𝑘,𝑙 |≤ 𝛿(𝑆), where 𝑝𝑒𝑞𝑆 { { { { if 2𝑁2𝑝 (𝑆2 ) ≤ 𝑒𝑝𝑞𝑆 , { { { −1/2𝑝 { 2𝑁2𝑝 (𝑆2 ) 𝛿 (𝑆) := { 2 { 2𝑁2𝑝 (𝑆 ) [ln ( )] { { 𝑞𝑆 𝑝 { { { { 2 { if 2𝑁2𝑝 (𝑆 ) > 𝑒𝑝𝑞𝑆 . (34) Journal of Complex Analysis 5 Let 𝑟𝑠 (𝑃2 ) be the spectral radius of 𝑃2 : 𝑟𝑠 (𝑃2 ) := sup𝑠∈Σ(𝑃2 ) |𝑠|. From the previous corollary, it follows that 𝑟𝑠2 (𝑃2 ) ≤ 𝑟𝑠2 (𝑄) + 𝛿(𝑆). Besides, 𝑟𝑠 (𝑄) = sup 𝑙=1,2; 𝑘=1,2,... 󵄨 󵄨󵄨 󵄨󵄨𝑧𝑘𝑙 (𝑄)󵄨󵄨󵄨 . (35) Corollary 12. Under condition (33), let Then, 𝑟𝑠 (𝑃2 ) < 1. 𝑟𝑠2 (𝑄) + 𝛿 (𝑆) < 1. (36) In the present section, we briefly discuss applications of our results to difference and differential equations. Consider the difference equation 𝑚 𝑘=0 (𝐴 0 = 𝐼; 𝑗 = 𝑚, 𝑚 + 1, . . .) , (37) with bounded operator coefficients 𝐴 𝑘 . This equation is said to be asymptotically stable, if any of its solution tends to zero as 𝑡 → ∞. It is not hard to check that (37) is asymptotically stable, provided 𝑟𝑠 (𝑃𝑚 ) < 1; (38) cf. [13]. Now one can use the perturbation results due to Theorems 3 and 4. For example, let 𝑚 = 2. So, V (𝑗 + 2) + 𝐴 1 V (𝑗 + 1) + 𝐴 2 V (𝑗) = 0 (𝑗 = 2, 3, . . .) . (39) Take an operator 𝐵 commuting with 𝐴 1 as in the previous section. Recall that it is desirable to choose 𝐵 such that the norm of 𝐴 2 − 𝐵 is small enough. Now Corollary 12 implies the following. Corollary 13. Under conditions (33) and (36), (39) is asymptotically stable. Furthermore, let us consider in 𝐻 the differential equation 𝑢𝑥𝑥 (𝑥) + 𝐴 1 𝑢𝑥 (𝑥) + 𝐴 2 𝑢 (𝑥) = 𝑓 (𝑥) where ∞ 𝑓 (𝑥) = ∑ 𝑓𝑘 𝑒2𝜋𝑖𝑘𝑥 ∈ 𝐿2 ([0, 1] , 𝐻) 𝑘=−∞ (0 < 𝑥 < 1) , (40) (𝑓𝑘 ∈ 𝐻) . (41) Numerous integrodifferential equations can be written in the form of (40). Impose the periodic conditions 𝑢󸀠 (0) = 𝑢󸀠 (1) . 𝑢 (0) = 𝑢 (1) , (42) We seek a solution of problem (40), (42) in the form ∞ 𝑢 (𝑥) = ∑ 𝑢𝑘 𝑒2𝜋𝑖𝑘𝑥 , 𝑘=−∞ (−4𝜋2 𝑘2 𝐼 + 2𝜋𝑖𝑘𝐴 1 + 𝐴 2 ) 𝑢𝑘 = 𝑓𝑘 . (44) This equation has a solution provided the spectrum of 𝑃2 (𝑧) does not contain the numbers 2𝜋𝑖𝑘. Now one can apply Corollary 11. References 5. Difference and Differential Equations ∑ 𝐴 𝑚−𝑘 V (𝑗 + 𝑘) = 0 where 𝑢𝑘 (𝑘 = 0, ±1, . . .) should be found. Substituting this expression into (40), we obtain (43) [1] L. Rodman, An Introduction to Operator Polynomials, vol. 38 of Operator Theory: Advances and Applications, Birkhäuser, Basel, Switzerland, 1989. [2] E. Bairamov, Ö. Çakar, and A. M. Krall, “Spectral properties, including spectral singularities, of a quadratic pencil of Schrödinger operators on the whole real axis,” Quaestiones Mathematicae, vol. 26, no. 1, pp. 15–30, 2003. [3] P. A. Cojuhari, “Estimates of the discrete spectrum of a linear operator pencil,” Journal of Mathematical Analysis and Applications, vol. 326, no. 2, pp. 1394–1409, 2007. [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. 1837–1849, 2011. [5] M. I. Gil’, “On bounds for spectra of operator pencils in a Hilbert space,” Acta Mathematica Sinica (English Series), vol. 19, no. 2, pp. 313–326, 2003. [6] M. I. Gil’, “Bounds for the spectrum of analytic quasinormal operator pencils,” Communications in Contemporary Mathematics, vol. 5, no. 1, pp. 101–118, 2003. [7] M. I. Gil’, “Sums of characteristic values of compact polynomial operator pencils,” Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 1469–1476, 2008. [8] M. Hasanov, “An approximation method in the variational theory of the spectrum of operator pencils,” Acta Applicandae Mathematicae, vol. 71, no. 2, pp. 117–126, 2002. [9] I. V. Kurbatova, “A Banach algebra associated with a linear operator pencil,” Matematicheskie Zametki, vol. 86, no. 3, pp. 394–401, 2009 (Russian), translation in Mathematical Notes, vol. 86 (2009), no. 3-4, 361–367. [10] I. V. Kurbatova, “A functional calculus generated by a quadratic operator pencil,” Journal of Mathematical Sciences, vol. 182, no. 5, pp. 646–655, 2012. [11] M. D. Manafov and A. Kablan, “On a quadratic pencil of differential operators with periodic generalized potential,” International Journal of Pure and Applied Mathematics, vol. 50, no. 4, pp. 515–522, 2009. [12] Y. Yakubov, “Fold completeness of a system of root vectors of a system of unbounded polynomial operator pencils in Banach spaces. I. Abstract theory,” Journal de Mathématiques Pures et Appliquées, vol. 92, no. 3, pp. 263–275, 2009. [13] M. I. Gil’, Difference Equations in Normed Spaces: Stability and Oscillations, vol. 206 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2007. [14] M. I. 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