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

Let $Ω$ be a subdomain of $\mathbb{C}$ and let $μ$ be a positive Borel measure on $Ω$. In this paper, we study the asymptotic behavior of the eigenvalues of compact Toeplitz operator $T_μ$ acting on Bergman spaces on $Ω$. Let $(λ_n(T_μ))$ be the decreasing sequence of the eigenvalues of $T_μ$ and let $ρ$ be an increasing function such that $ρ(n)/n^A$ is decreasing for some $A>0$. We give an explicit necessary and sufficient geometric condition on $μ$ in order to have $λ_n(T_μ)\asymp 1/ρ(n)$. As applications, we consider composition operators $C_φ$, acting on some standard analytic spaces on the unit disc $\mathbb{D}$. First, we give a general criterion ensuring that the singular values of $C_φ$ satisfy $s_n(C_φ) \asymp 1/ρ(n)$. Next, we focus our attention on composition operators with univalent symbols, where we express our general criterion in terms of the harmonic measure of $φ\mathbb{D})$. We finally study the case where $\partial φ(\mathbb{D})$ 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 $h(T_μ)$, where $h$ is suitable concave or convex functions.

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