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 \(L^2_a(D)\) 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.