Non-Existence of Subordinate Solutions for Jacobi Operators in Some Critical Cases

Moszyński, Marcin

doi:10.1007/s00020-012-2018-0

Cited by 3 publications

(6 citation statements)

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“…Precise description of this problem and its solution (in our setting) the reader will find in the next section. In [9] M. Moszyński investigates this problem in general situation.…”

Section: Resultsmentioning

confidence: 99%

“…This kind of situation was studied by M.Moszyński in [9]. Unfortunately particular results of his paper do not cover Jacobi operators considered in this paper.…”

Section: Wojciech Motykamentioning

confidence: 95%

“…Such operators have been studied e.g. in [3,7,[9][10][11]. In all of the papers spectral properties of the considered Jacobi operators were obtained in the same way: the asymptotic behavior of the generalized eigenvectors + the subordinacy theory.…”

Section: Introductionmentioning

confidence: 99%

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Spectra of some selfadjoint Jacobi operators in the double root case

Motyka¹

2015

OpMath

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“…Precise description of this problem and its solution (in our setting) the reader will find in the next section. In [9] M. Moszyński investigates this problem in general situation.…”

Section: Resultsmentioning

confidence: 99%

“…This kind of situation was studied by M.Moszyński in [9]. Unfortunately particular results of his paper do not cover Jacobi operators considered in this paper.…”

Section: Wojciech Motykamentioning

confidence: 95%

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Spectra of some selfadjoint Jacobi operators in the double root case

Motyka¹

2015

OpMath

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“…Different spectral results could be related to different kinds of asymptotic properties. And for some asymptotic properties the use of 2d-generalized eigenvectors can be more suitable than the use of generalized eigenvectorssee, e.g., [10]. These methods, however, are less developed in the case of block Jacobi operators (with d > 1).…”

Section: Jacobi Operators: Generalized Eigenvectorsmentioning

confidence: 99%

“…The name C 2 -vector-generalised eigenvector was also used for d = 1-see, e.g.,[10] 3 Note that it is equall to the minimal s-number of A. And it is not a norm in B(X), if dim X > 1.…”

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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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Orthogonal polynomials with periodically modulated recurrence coefficients in the Jordan block case

Świderski¹,

Trojan²

2020

Preprint

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GRZEGORZ ŚWIDERSKI AND BARTOSZ TROJAN A. We study orthogonal polynomials with periodically modulated entries when 0 lies on the hard edge of the spectrum of the corresponding periodic Jacobi matrix. In particular, we show that their orthogonality measure is purely absolutely continuous on a real half-line and purely discrete on its complement. Additionally, we provide the constructive formula for density in terms of Turán determinants. Moreover, we determine the exact asymptotic behavior of the orthogonal polynomials. Finally, we study scaling limits of the Christoffel-Darboux kernel.

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Non-Existence of Subordinate Solutions for Jacobi Operators in Some Critical Cases

Cited by 3 publications

References 30 publications

Spectra of some selfadjoint Jacobi operators in the double root case

Spectra of some selfadjoint Jacobi operators in the double root case

Uni-asymptotic Linear systems and Jacobi Operators

Orthogonal polynomials with periodically modulated recurrence coefficients in the Jordan block case

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