Compact multipliers on spaces of analytic functions

TL;DR

This paper characterizes when multiplier operators on Banach spaces of analytic functions are compact, providing necessary and sufficient conditions based on the space embeddings and measures of noncompactness.

Contribution

It offers new criteria for compactness of multipliers on analytic function spaces, especially within the embeddings involving $H_$ and $H_$, and computes measures of noncompactness for diagonal operators.

Findings

Necessary and sufficient conditions for compactness when $H_ \u2192 X \u2192 H_$.

Characterization of compact multipliers for $H_ \u2192 X \u2192 H_$.

Calculation of Hausdorff measure of noncompactness for diagonal operators.

Abstract

In the paper compact multiplier operators on Banach spaces of analytic functions on the unit disk with the range in Banach sequence lattices are studied. If the domain space $X$ is such that $H_\infty\hookrightarrow X\hookrightarrow H_1$, necessary and sufficient conditions for compactness are presented. Moreover, the calculation of the Hausdorff measure of noncompactness for diagonal operators between Banach sequence lattices is applied to obtaining the characterization of compact multipliers in case the domain space $X$ satisfies $H_\infty\hookrightarrow X\hookrightarrow H_2$.