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1

Spectral resolutions for non-self-adjoint block convolution operators

Zalot Ewelina

Opuscula Mathematica

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2022

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Vol. 42, no. 3

459--487

EN

The paper concerns the spectral theory for a class of non-self-adjoint block convolution operators. We mainly discuss the spectral representations of such operators. It is considered the general case of operators defined on Banach spaces. The main results are applied to periodic Jacobi matrices.

2

On one condition of absolutely continuous spectrum for self-adjoint operators and its applications

Ianovich E.

Opuscula Mathematica

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2018

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Vol. 38, no. 5

699--718

EN

In this work the method of analyzing of the absolutely continuous spectrum for self-adjoint operators is considered. For the analysis it is used an approximation of a self-adjoint operator A by a sequence of operators An with absolutely continuous spectrum on a given interval [a, b] which converges to A in a strong sense on a dense set. The notion of equi-absolute continuity is also used. It was found a sufficient condition of absolute continuity of the operator A spectrum on the finite interval [a, b] and the condition for that the corresponding spectral density belongs to the class Lp[a,b] (p ≥ 1). The application of this method to Jacobi matrices is considered. As one of the results we obtain the following assertion: Under some mild assumptions, suppose that there exist a constant C > 0 and a positive function g(x) ∈ Lp[a, b] (p ≥ 1).such that for all n sufficiently large and almost all [formula] the estimate [formula] holds, where Pn(x) are 1st type polynomials associated with Jacobi matrix (in the sense of Akhiezer) and bn is a second diagonal sequence of Jacobi matrix. Then the spectrum of Jacobi matrix operator is purely absolutely continuous on [a, b] and for the corresponding spectral density ƒ (x) we have ƒ (x) ∈ Lp[a,b].

3

Spectra of some selfadjoint Jacobi operators in the double root case

Motyka W.

Opuscula Mathematica

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2015

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Vol. 35, no. 3

353--370

EN

In this paper we prove a mixed spectrum of Jacobi operators defined by λn = s(n)(1 + x(n)) and qn = — 2s(n)(l+y/(n)), where (s(n)) is a real unbounded sequence, (x(n)) and (y(n)) are some perturbations.

4

Spectral representations for a class of banded Jacobi-type matrices

Zalot E., Majdak W.

Opuscula Mathematica

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2014

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Vol. 34, no. 4

871--887

EN

We describe some spectral representations for a class of non-self-adjoint banded Jacobi-type matrices. Our results extend those obtained by P.B. Naïman for (two-sided infinite) periodic tridiagonal Jacobi matrices.

5

Weyl-Titchmarsh type formula for Hermite operator with small perturbation

Simonov S.

Opuscula Mathematica

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2009

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Vol. 29, no. 2

187-207

EN

Small perturbations of the Jacobi matrix with weights √n and zero diagonal are considered. A formula relating the asymptotics of polynomials of the first kind to the spectral density is obtained, which is an analogue of the classical Weyl-Titchmarsh formula for the Schr ödinger operator on the half-line with summable potential. Additionally, a base of generalized eigenvectors for "free" Hermite operator is studied and asymptotics of Plancherel-Rotach type are obtained.

6

A first-order spectral phase transition in a class of periodically modulated Hermitian Jacobi matrices

Pchelintseva I.

Opuscula Mathematica

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2008

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Vol. 28, no. 2

137-150

EN

We consider self-adjoint unbounded Jacobi matrices with diagonal q(n) = b(n)n and off-diagonal entries λ(n) = n, where b(n) is a 2-periodical sequence of real numbers. The parameter space is decomposed into several separate regions, where the spectrum of the operator is either purely absolutely continuous or discrete. We study the situation where the spectral phase transition occurs, namely the case of b(1)b(2) = 4. The main motive of the paper is the investigation of asymptotics of generalized eigenvectors of the Jacobi matrix. The pure point part of the spectrum is analyzed in detail.

7

Unbounded Jacobi matrices with empty absolutely continuous spectrum

Cojuhari P., Janas J.

Bulletin of the Polish Academy of Sciences. Mathematics

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2008

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Vol. 56, no 1

39-51

EN

Sufficient conditions for the absence of absolutely continuous spectrum for unbounded Jacobi operators are given. A class of unbounded Jacobi operators with purely singular continuous spectrum is constructed as well.