On 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 - UMR 8524
3 Departamento de Analisis Matematico

We show that the approximation numbers of a compact composition operator on the weighted Bergman spaces $\mathfrak{B}_\alpha$ of the unit disk can tend to 0 arbitrarily slowly, but that they never tend quickly to 0: they grow at least exponentially, and this speed of convergence is only obtained for symbols which do not approach the unit circle. We also give an upper bounds and explicit an example.

Mots clés: approximation number; Bergman space; Carleson measure; composition operator; Hardy space; interpolation sequence; reproducing kernel; weighted Bergman space; weighted shift

Domaines: Analyse fonctionnelle [math.FA]

Daniel Li, Hervé Queffélec, Luis Rodriguez-Piazza. On approximation numbers of composition operators. 2011. ⟨hal-00588256⟩