Composition Operators on the Bloch space of the Unit Ball of a Hilbert Space

TL;DR

This paper characterizes when composition operators induced by analytic self-maps of the unit ball in a Hilbert space are compact on the Bloch space, providing necessary and sufficient conditions and illustrative examples.

Contribution

It establishes precise criteria for the compactness of composition operators on the Bloch space in a Hilbert space setting, extending previous understanding.

Findings

Characterization of compactness conditions for composition operators

Examples illustrating the connection between conditions and compactness

Extension of Bloch space operator theory to Hilbert space unit balls

Abstract

Every analytic self-map of the unit ball of a Hilbert space induces a bounded composition operator on the space of Bloch functions. Necessary and sufficient conditions for compactness of such composition operators are provided, as well as some examples that clarify the connections among such conditions.