Some new properties of composition operators associated with lens maps

TL;DR

This paper explores new properties of composition operators linked to lens maps, focusing on approximation numbers in Hardy spaces and the compactness behavior in Hardy-Orlicz and Bergman-Orlicz spaces, revealing negative answers to existing questions.

Contribution

It provides novel examples and results on the approximation numbers and compactness properties of composition operators associated with lens maps, especially in non-reflexive spaces.

Findings

Approximation numbers of composition operators on Hardy space $H^2$ are characterized.

Counterexamples show not all weakly compact composition operators are norm-compact in non-reflexive spaces.

Negative answer to whether all weakly compact operators are norm-compact in certain Orlicz spaces.

Abstract

We give examples of results on composition operators connected with lens maps. The first two concern the approximation numbers of those operators acting on the usual Hardy space $H^2$. The last ones are connected with Hardy-Orlicz and Bergman-Orlicz spaces $H^\psi$ and $B^\psi$, and provide a negative answer to the question of knowing if all composition operators which are weakly compact on a non-reflexive space are norm-compact.