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

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.