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

Let $L^2(D)$ be the space of measurable square-summable functions on the unit disk. Let $L_a^2(D)$ 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 $L_a^2(D)$. For a function $\phi \in L^{\mathrm{\infty }}(D)$, the Toeplitz operator $T_{\phi }:L_a^2(D)\to L_a^2(D)$ is defined as $$T_{\phi }f=P_{+}\phi f,f\in L_a^2(D)\mathrm{.}$$ The main result of this article are spectral asymptotics for singular (or eigen-) values of compact Toeplitz operators with logarithmically decaying symbols, that is $$\phi (z)={\phi }_1(e^{i\theta })(1+\log (1\mathrm{/}(1-r)){)}^{-\gamma },\gamma >0,$$ where $z=r{e}^{i\theta }$ and ${\phi }_1$ 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.