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

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.