A spectral radius type formula for approximation numbers of composition operators

Abstract

For approximation numbers $a_n(C_{\ddot{I}}•)$ of composition operators $C_{\ddot{I}}•$ on weighted analytic Hilbert spaces, including the Hardy, Bergman and Dirichlet cases, with symbol $\ddot{I}•$ of uniform norm $<1$, we prove that $\lim_{n \to \infty} [a_n (C_ϕ)]^{1/n} = \e^{- 1/ \capa [ϕ(\D)]}$, where $\capa [ϕ(\D)]$ is the Green capacity of $ϕ(\D)$ in $\D$. This formula holds also for $H^p$ with $1\le p<\infty$.

25 pages