Compact and weakly compact composition operators from the Bloch space into M\"obius invariant spaces

TL;DR

This paper provides a comprehensive analysis of the boundedness, compactness, and weak compactness of composition operators from the Bloch space into various conformally invariant spaces, establishing key equivalences among these properties.

Contribution

It offers a unified approach to characterize these operators across multiple conformally invariant spaces, demonstrating that weak compactness coincides with compactness in this context.

Findings

Weak compactness is equivalent to compactness for these operators.

Boundedness and compactness are equivalent for operators into the small spaces.

Provides unified criteria for boundedness and compactness across various spaces.

Abstract

We obtain exhaustive results and treat in a unified way the question of boundedness, compactness, and weak compactness of composition operators from the Bloch space into any space from a large family of conformally invariant spaces that includes the classical spaces like $BMOA$, $Q_\alpha$, and analytic Besov spaces $B^p$. In particular, by combining techniques from both complex and functional analysis, we prove that in this setting weak compactness is equivalent to compactness. For the operators into the corresponding "small" spaces we also characterize the boundedness and show that it is equivalent to compactness.