A spectral radius type formula for approximation numbers of composition operators

TL;DR

This paper establishes a spectral radius formula for the approximation numbers of composition operators on various weighted analytic Hilbert spaces, linking their asymptotic behavior to the Green capacity of the image of the unit disk.

Contribution

It introduces a spectral radius type formula for approximation numbers of composition operators, connecting their decay rate to Green capacity, applicable across multiple function spaces.

Findings

Asymptotic formula for approximation numbers involving Green capacity

Extension of the formula to Hardy spaces $H^p$ for $1 \,\leq\, p < \infty$

Unified approach for weighted analytic Hilbert spaces including Hardy, Bergman, and Dirichlet spaces

Abstract

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