Let $L^2({{\mathbb{D}}})$ be the space of measurable square-summable functions on the unit disk. Let $L^2_a({{\mathbb{D}}})$ be the Bergman space, that is, the (closed) subspace of analytic functions in $L^2({{\mathbb{D}}})$. $P_+$ stays for the orthogonal projection going from $L^2({{\mathbb{D}}})$ to $L^2_a({{\mathbb{D}}})$. For a function $\varphi \in L^\infty ({{\mathbb{D}}})$, the Toeplitz operator $T_\varphi : L^2_a({{\mathbb{D}}})\to L^2_a({{\mathbb{D}}})$ is defined as $$\begin{align*} & T_\varphi f=P_+\varphi f, \quad f\in L^2_a({{\mathbb{D}}}). \end{align*}$$The main result of this article are spectral asymptotics for singular (or eigen-) values of compact Toeplitz operators with logarithmically decaying symbols, that is, $$\begin{align*} & \varphi(z)=\varphi_1(e^{i\theta})\, (1+\log(1/(1-r)))^{-\gamma},\quad \gamma>0, \end{align*}$$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.
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 ϕ ∈ L ∞ (D), the Toeplitz operatorThe main result of this article are spectral asymptotics for singular (or eigen-) values of compact Toeplitz operators with logarithmically decaying symbols, that is ϕ(z) = ϕ 1 (e iθ ) (1 + log(1/(1 − r))) −γ , γ > 0, where z = re iθ and ϕ 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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.