Composition operators on vector-valued analytic function spaces: a survey

TL;DR

This survey reviews recent research on composition operators acting on vector-valued analytic function spaces, focusing on qualitative properties like weak compactness and including new insights on Hardy and BMOA spaces.

Contribution

It provides a comprehensive overview of the current state of research on composition operators on vector-valued analytic spaces, highlighting recent results and open problems.

Findings

Analysis of weak compactness properties of composition operators

New observations on Hardy and BMOA vector-valued spaces

Discussion of open problems in the field

Abstract

We survey recent results about composition operators induced by analytic self-maps of the unit disk in the complex plane on various Banach spaces of analytic functions taking values in infinite-dimensional Banach spaces. We mostly concentrate on the research line into qualitative properties such as weak compactness, initiated by Liu, Saksman and Tylli (1998), and continued in several other papers. We discuss composition operators on strong, respectively weak, spaces of vector-valued analytic functions, as well as between weak and strong spaces. As concrete examples, we review more carefully and present some new observations in the cases of vector-valued Hardy and BMOA spaces, though the study of composition operators has been extended to a wide range of spaces of vector-valued analytic functions, including spaces defined on other domains. Several open problems are stated.