Compactness and related properties of weighted composition operators on weighted BMOA spaces

TL;DR

This paper investigates the properties of weighted composition operators on various weighted BMOA and VMOA spaces, establishing equivalences among properties like compactness and weak compactness, and providing new characterizations and test functions.

Contribution

It extends previous theorems to weighted spaces, introduces new test functions, and characterizes properties like compactness for weighted composition operators.

Findings

Properties such as compactness and weak compactness coincide for many weighted BMOA and VMOA spaces.

New function-theoretic characterizations for these properties are provided.

The results generalize earlier theorems and include new test functions suitable for weighted settings.

Abstract

It is shown that a large class of properties coincide for weighted composition operators on a large class of weighted VMOA spaces, including the ones with logarithmic weights and the ones with standard weights $(1-|z|)^{-c}, \ 0\leq c< 1/2$. Some of these properties are compactness, weak compactness, complete continuity and strict singularity. A function-theoretic characterization for these properties is also given. Similar results are also proved for many weighted composition operators on similarly weighted BMOA spaces. The main results extend the theorems given in [Proc. Amer. Math. Soc. 151 (2023), 1195--1207], and new test functions that are suitable for the weighted setting are developed.