Spectral Properties of Block Jacobi Matrices

Świderski, Grzegorz

doi:10.1007/s00365-018-9420-z

Cited by 6 publications

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“…The study of the sequence (S γ n : n ∈ N) is motivated by the following theorem, whose proof is analogous to the proof of [15,Theorem 7]. We include it for the sake of completeness.…”

Section: Definitions and Basic Propertiesmentioning

confidence: 99%

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Spectral Properties of Some Complex Jacobi Matrices

Świderski

2020

Integr. Equ. Oper. Theory

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A. We study spectral properties of bounded and unbounded complex Jacobi matrices. In particular, we formulate conditions assuring that the spectrum of the studied operators is continuous on some subsets of the complex plane and we provide uniform asymptotics of their generalised eigenvectors. We illustrate our results by considering complex perturbations of real Jacobi matrices belonging to several classes: asymptotically periodic, periodically modulated and the blend of these two. Moreover, we provide conditions implying existence of a unique closed extension. The method of the proof is based on the analysis of a generalisation of shifted Turán determinants to the complex setting.2010 Mathematics Subject Classification. Primary: 47B36.

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“…The study of the sequence (S γ n : n ∈ N) is motivated by the following theorem, whose proof is analogous to the proof of [15,Theorem 7]. We include it for the sake of completeness.…”

Section: Definitions and Basic Propertiesmentioning

confidence: 99%

“…This idea seems to be difficult to apply in our setting. Instead we extend some ideas from our recent articles [15,16]. The structure of the article is as follows.…”

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confidence: 99%

Spectral Properties of Some Complex Jacobi Matrices

Świderski

2020

Integr. Equ. Oper. Theory

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“…and 2d -generalized Eigenvectors Let us start from recalling the notions of block Jacobi matrix and operator (see, e.g., [2,6,13] and citations therein) and its particular "scalar" case. Let 1 and let A 1 , B 1 , A 2 , B 2 , .…”

Section: Jacobi Operators: Generalized Eigenvectorsmentioning

confidence: 99%

“…They contain very interesting approach, leading in particular also to uni-asymptoticity results for 2-generalized eigenvectors in scalar case (together with some extra "λ-uniform" properties)-see [12]. The paper [13] provides some spectral results, including also some essential spectrum results for the block case. Instead of a construction of Weyl sequences, the method is based on checking that there is no any generalized eigenvector in 2 (N, C d ).…”

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Uni-asymptotic Linear systems and Jacobi Operators

Moszyński

2019

Integr. Equ. Oper. Theory

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A family {Us} s∈S of bounded linear operators in a normed space X is uni-asymptotic, when all its trajectories {Usx} s∈S with x = 0 have the same norm-asymptotic behavior (see 1.5); {Us} s∈S is tight, when the operator norm and the minimal modulus of Us have the same asymptotic behavior (see 1.6). We prove that uni-asymptoticity is equivalent to tightness if dim X < +∞, and that the finite dimension is essential. Some other conditions equivalent to uni-asymptoticity are provided, including asymptotic formulae for the operator norm and for the trajectories, expressed in terms of determinants det Us (see Theorem 1.7). We find a connection of these abstract results with some results and notions from spectral theory of Jacobi operators, e.g., with the H-class property for transfer matrix sequence.Mathematics Subject Classification. 47D06, 47B36, 81Q10.1 In this paper we identify indexed family {fs} s∈S with the function on S given by: S s −→ fs. 23 Page 2 of 15 M. Moszyński IEOTAbove S represents the set of "admissible moments of time", and X-the set of "admissible states" for the process. Hence, in this paper, for fixed x ∈ X, we use the name trajectory (or orbit), when we consider the family {U s x} s∈S of elements of X, obtained by a family {U s } s∈S of operators. Let us stress, however, that the actual role of the mathematical objects X, S, {U s } s∈S and {U s x} s∈S can be essentially different than their typical role as above-that is, as in mathematical descriptions of some real processes, e.g, in physics etc. As we shall see in Sect. 2 (see 2.28), those objects can be also useful for mathematical description of some "purely mathematical" processes for purely mathematical goals.In this paper we consider only the linear case, i.e., X is a Banach space here and all the operators U s are linear (and we shall mainly assume finite dimension of X). However, some notions introduced here have also sense in non-linear case, and they could be worth future research.Section 1 is devoted to some abstract studies of so-called uni-asymptotic families of operators, i.e., such families {U s } s∈S that all its non-zero trajectories {U s x} s∈S have "the same norm-asymptotic behavior". This precisely means, that for any non-zero initial conditions x, y ∈ X we have3). Although the word "asymptotic" may not fit well to the above definition for the general S, it makes sense for some ordered S-s, e.g., for S = N n0 (see 0.1). The name "uni-asymptotic" is used also for linear spaces consisting of some functions f = {f (s)} s∈S from S into X, with a natural analogic meaning. This section contains the main results of the paper-Theorems 1.5 and 1.7 on some equivalent conditions for uni-asymptoticity in finite dimension. One of the most important observations is that the uni-asymptoticity is equivalent to tightness, where {U s } s∈S is tight, when the operator norm and the minimal modulus of U s have the same asymptotic behavior (see 1.6). Examples showing the necessity of the dim X < ∞ assumption are also provided. They show as well, th...

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