Two remarks on composition operators on the Dirichlet space

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

We show that the decay of approximation numbers of compact composition operators on the Dirichlet space $\mathcal{D}$ can be as slow as we wish, which was left open in the cited work. We also prove the optimality of a result of O.~El-Fallah, K.~Kellay, M.~Shabankhah and A.~Youssfi on boundedness on $\mathcal{D}$ of self-maps of the disk all of whose powers are norm-bounded in $\mathcal{D}$.

Keywords: approximation numbers; Carleson embedding; composition operator; cusp map; Dirichlet space

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

Daniel Li, Hervé Queffélec, Luis Rodriguez-Piazza. Two remarks on composition operators on the Dirichlet space. 2014. ⟨hal-00982257v2⟩