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

Omar El-Fallah, Mohamed El Ibbaoui

• Let Ω be a subdomain of C and let μ be a positive Borel measure on ΩΩ. In this paper, we study the asymptotic behavior of the eigenvalues of compact Toeplitz operators Tμ acting on Bergman spaces on ΩΩ. Let (λn(Tμ)) be the decreasing sequence of the eigenvalues of Tμ, and let ρ be an increasing function such that ρ(n)/nA is decreasing for some >0A>0. We give an explicit necessary and sufficient geometric condition on μ in order to have λn(Tμ)≍1/ρ(n). As applications, we consider composition operators Cφ, acting on some standard analytic spaces on the unit disc D. First, we give a general criterion ensuring that the singular values of Cφ satisfy sn(Cφ)≍1/ρ(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 φ(D). We finally study the case where ∂φ(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(Tμ), where ℎh is a suitable concave or convex function