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

Let be a subdomain of C and let be a positive Borel measure on. In this paper, we study the asymptotic behavior of the eigenvalues of compact Toeplitz operators acting on Bergman spaces on. Let. T be the decreasing sequence of the eigenvalues of T, and let be an increasing function such that .n/=nA is decreasing for some A > 0. We give an explicit necessary and sufficient geometric condition on in order to have As applications, we consider composition operators C', acting on some standard analytic spaces on the unit disc D. First, we give a general criterion ensuring that the singular values of C' satisfy sn. Next, we focus our attention on composition operators with univalent symbols, where we express our general criterion in terms of the harmonic measure of '.. We finally study the case where @'. 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. where h is a suitable concave or convex function. [ABSTRACT FROM AUTHOR]