A spectral radius type formula for approximation numbers of composition operators

Daniel Li (1) , Hervé Queffélec (2) , Luis Rodriguez-Piazza (3)

1 LML - Laboratoire de Mathématiques de Lens
2 LPP - Laboratoire Paul Painlevé
3 Departamento de Analisis Matematico

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 $1 \leq p < \infty$.

Keywords: approximation numbers; Bergman space; composition operator; Dirichlet space; Green capacity; Hardy space; weighted analytic Hilbert space

Domains: Mathematics [math]; Functional Analysis [math.FA]

Daniel Li, Hervé Queffélec, Luis Rodriguez-Piazza. A spectral radius type formula for approximation numbers of composition operators. 2014. ⟨hal-01020782⟩