Approximation numbers of composition operators on the Dirichlet space

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

1 LML - Laboratoire de Mathématiques de Lens
2 departamento analisis matematico
3 LPP - Laboratoire Paul Painlevé - UMR 8524

We study the decay of approximation numbers of compact composition operators on the Dirichlet space. We give upper and lower bounds for these numbers. In particular, we improve on a result of O. El-Fallah, K. Kellay, M. Shabankhah and A. Youssfi, on the set of contact points with the unit circle of a compact symbolic composition operator acting on the Dirichlet space D. We extend their results in two directions: first, the contact only takes place at the point 1. Moreover, the approximation numbers of the operator can be arbitrarily sub-exponentially small.

Mots clés: approximation numbers; capacit; composition operator; cusp map; Dirichlet space; Schatten classes

Domaines: Analyse fonctionnelle [math.FA]

Pascal Lefèvre, Daniel Li, Luis Rodriguez-Piazza, Hervé Queffélec. Approximation numbers of composition operators on the Dirichlet space. 2012. ⟨hal-00766018⟩