On spectral properties of compact Toeplitz operators on Bergman space with logarithmically decaying symbol and applications to banded matrices

TL;DR

This paper investigates the spectral asymptotics of compact Toeplitz operators on Bergman space with symbols that decay logarithmically, and applies these results to analyze the spectra of banded matrices.

Contribution

It provides new spectral asymptotics for Toeplitz operators with logarithmically decaying symbols, extending understanding of their eigenvalue behavior and applications to banded matrices.

Findings

Spectral asymptotics for eigenvalues of Toeplitz operators with logarithmic decay.

Application of spectral results to banded and Jacobi matrices.

Characterization of eigenvalue decay rates for specific symbol classes.

Abstract

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