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Let $L^2(D)$ be the space of measurable square-summable functions on the unitdisk. Let $L^2_a(D)$ be the Bergman space, i.e., the (closed) subspace ofanalytic functions in $L^2(D)$. $P_+$ stays for the orthogonal projection goingfrom $L^2(D)$ to $L^2_a(D)$. For a function $\varphi\in L^\infty(D)$, theToeplitz operator $T_\varphi: L^2_a(D)\to L^2_a(D)$ is defined as $$ T_\varphif=P_+\varphi f, \quad f\in L^2_a(D). $$ The main result of this article arespectral asymptotics for singular (or eigen-) values of compact Toeplitzoperators 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-wisecontinuous) function on the unit circle. The result is applied to the spectralanalysis of banded (including Jacobi) matrices.

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