Trace estimates of Toeplitz operators on Bergman spaces and applications to composition operators (Q2104847)

Summary: 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 \(C_\varphi\), 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 a suitable concave or convex function.