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

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 operators $T_{\mu }$ 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 $C_{\phi }$, acting on some standard analytic spaces on the unit disc $\mathbb{D}$. First, we give a general criterion ensuring that the singular values of $C_{\phi }$ satisfy $s_n(C_{\phi })\asymp 1/\rho (n)$. Next, we focus our attention on composition operators with univalent symbols, where we express our general criterion in terms of the harmonic measure of $\phi (\mathbb{D})$. We finally study the case where $\partial \phi (\mathbb{D})$ 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 a suitable concave or convex function.