Spectral Properties of Jacobi Matrices with Rapidly Growing Power-like Weights

2002 ◽  
pp. 387-394 ◽  
Author(s):  
Luis O. Silva
Keyword(s):  
Spectral Properties ◽  
Jacobi Matrices

scholarly journals On Spectral Properties of Compact Toeplitz Operators on Bergman Space with Logarithmically Decaying Symbol and Applications to Banded Matrices

2021 ◽  
Author(s):  
Mahamet Koïta ◽  
Stanislas Kupin ◽  
Sergey Naboko ◽  
Belco Touré
Keyword(s):  
Spectral Analysis ◽  
Continuous Function ◽  
Bergman Space ◽  
Orthogonal Projection ◽  
Spectral Properties ◽  
Toeplitz Operators ◽  
Closed Subspace ◽  
Jacobi Matrices ◽  
Banded Matrices ◽  
Eigen Values

Abstract Let $L^2({{\mathbb{D}}})$ be the space of measurable square-summable functions on the unit disk. Let $L^2_a({{\mathbb{D}}})$ be the Bergman space, that is, the (closed) subspace of analytic functions in $L^2({{\mathbb{D}}})$. $P_+$ stays for the orthogonal projection going from $L^2({{\mathbb{D}}})$ to $L^2_a({{\mathbb{D}}})$. For a function $\varphi \in L^\infty ({{\mathbb{D}}})$, the Toeplitz operator $T_\varphi : L^2_a({{\mathbb{D}}})\to L^2_a({{\mathbb{D}}})$ is defined as $$\begin{align*} & T_\varphi f=P_+\varphi f, \quad f\in L^2_a({{\mathbb{D}}}). \end{align*}$$The main result of this article are spectral asymptotics for singular (or eigen-) values of compact Toeplitz operators with logarithmically decaying symbols, that is, $$\begin{align*} & \varphi(z)=\varphi_1(e^{i\theta})\, (1+\log(1/(1-r)))^{-\gamma},\quad \gamma>0, \end{align*}$$where $z=re^{i\theta }$ and $\varphi _1$ is a continuous (or piece-wise continuous) function on the unit circle. The result is applied to the spectral analysis of banded (including Jacobi) matrices.

scholarly journals Spectral Properties of Block Jacobi Matrices

2018 ◽  
Vol 48 (2) ◽  
pp. 301-335 ◽  
Author(s):  
Grzegorz Świderski
Keyword(s):  
Spectral Properties ◽  
Jacobi Matrices

scholarly journals A Formula for Eigenvalues of Jacobi Matrices with a Reflection Symmetry

2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
S. B. Rutkevich
Keyword(s):  
Spectral Properties ◽  
Jacobi Operator ◽  
Scattering Data ◽  
Jacobi Matrices ◽  
Reflection Symmetry ◽  
Matrix Elements ◽  
Discrete Schrödinger Operator ◽  
Infinite Dimensional ◽  
Jacobi Operators ◽  
Secondary Diagonal

The spectral properties of two special classes of Jacobi operators are studied. For the first class represented by the 2M-dimensional real Jacobi matrices whose entries are symmetric with respect to the secondary diagonal, a new polynomial identity relating the eigenvalues of such matrices with their matrix entries is obtained. In the limit M→∞ this identity induces some requirements, which should satisfy the scattering data of the resulting infinite-dimensional Jacobi operator in the half-line, of which super- and subdiagonal matrix elements are equal to -1. We obtain such requirements in the simplest case of the discrete Schrödinger operator acting in l2(N), which does not have bound and semibound states and whose potential has a compact support.

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