Compact composition operators on Hardy-Orlicz and weighted Bergman-Orlicz spaces on the ball

TL;DR

This paper characterizes the compactness of composition operators on Hardy-Orlicz and weighted Bergman-Orlicz spaces on the ball, revealing conditions under which these operators are compact and extending classical results.

Contribution

It provides new criteria for compactness of composition operators on Orlicz spaces, including a simplified proof of a classical result and insights into the influence of Orlicz function growth.

Findings

Compactness on all Hardy-Orlicz spaces implies compactness on H^{ \infty}

Existence of non-compact composition operators with maps into Koranyi regions for fast-growing Orlicz functions

Compactness criteria expressed via Orlicz 'Angular derivative'-type condition

Abstract

Using recent characterizations of the compactness of composition operators on Hardy-Orlicz and Bergman-Orlicz spaces on the ball, we first show that a composition operator which is compact on every Hardy-Orlicz (or Bergman-Orlicz) space has to be compact on H^{\infty}. Then, we prove that, for each Koranyi region \Gamma, there exists a map \phi taking the ball into \Gamma such that, C_{\phi} is not compact on H^{\psi}\left(\mathbb{B}_{N}\right), when \psi grows fast. Finally, we give another characterization of the compactness of composition operator on weighted Bergman-Orlicz spaces in terms of an Orlicz "Angular derivative"-type condition. This extends (and simplify the proof of) a result by K. Zhu for the classical Bergman case. Moreover, we deduce that the compactness of composition operators on weighted Bergman-Orlicz spaces does not depend on the weight anymore, when the Orlicz function grows fast.