Trace estimates of Toeplitz operators on Bergman spaces and applications to composition operators

TL;DR

This paper investigates the eigenvalue decay of Toeplitz operators on Bergman spaces and applies these results to characterize the singular values of composition operators, providing geometric criteria and concrete examples.

Contribution

It establishes explicit geometric conditions for eigenvalue asymptotics of Toeplitz operators and extends these results to composition operators with univalent symbols on the unit disk.

Findings

Eigenvalues of Toeplitz operators decay as 1/ρ(n) under geometric conditions.

General criterion for composition operators' singular values to decay as 1/ρ(n).

Explicit examples with boundary contact points of the symbol.

Abstract

Let $\Omega$ be a subdomain of $\mathbb{C}$ and let $\mu$ be a positive Borel measure on $\Omega$. In this paper, we study the asymptotic behavior of the eigenvalues of compact Toeplitz operator $T_\mu$ acting on Bergman spaces on $\Omega$. Let $(\lambda_n(T_\mu))$ be the decreasing sequence of the eigenvalues of $T_\mu$ and let $\rho$ be an increasing function such that $\rho (n)/n^A$ is decreasing for some $A>0$. We give an explicit necessary and sufficient geometric condition on $\mu$ in order to have $\lambda_n(T_\mu)\asymp 1/\rho (n)$. As applications, we consider composition operators $C_\varphi$, acting on some standard analytic spaces on the unit disc $\mathbb{D}$. First, we give a general criterion ensuring that the singular values of $C_\varphi$ satisfy $s_n(C_\varphi ) \asymp 1/\rho(n)$. Next, we focus our attention on composition operators with univalent symbols, where we express our general criterion in terms of the harmonic measure of $\varphi \mathbb{D})$. We finally study the case where $\partial \varphi (\mathbb{D})$ meets the unit circle in one point and give several concrete examples. Our method is based on upper and lower estimates of the trace of $h(T_\mu)$, where $h$ is suitable concave or convex functions.