Compact composition operators on Bergman-Orlicz spaces

TL;DR

This paper constructs specific composition operators that are compact on Hardy-Orlicz spaces but not on Bergman-Orlicz spaces, providing new characterizations of their compactness via Carleson functions and Nevanlinna counting functions.

Contribution

It introduces a novel construction of composition operators with differing compactness properties on Hardy- and Bergman-Orlicz spaces, and characterizes their compactness using Carleson and Nevanlinna functions.

Findings

Constructed an analytic self-map and Orlicz function with differing compactness properties.

Proved a Carleson embedding theorem for Bergman-Orlicz spaces.

Characterized compactness of composition operators via Carleson and Nevanlinna functions.

Abstract

We construct an analytic self-map $\phi$ of the unit disk and an Orlicz function $\Psi$ for which the composition operator of symbol $\phi$ is compact on the Hardy-Orlicz space $H^\Psi$, but not compact on the Bergman-Orlicz space ${\mathfrak B}^\Psi$. For that, we first prove a Carleson embedding theorem, and then characterize the compactness of composition operators on Bergman-Orlicz spaces, in terms of Carleson function (of order 2). We show that this Carleson function is equivalent to the Nevanlinna counting function of order 2.