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

David Norrbo

TL;DR

This work advances a comprehensive theory for weighted composition operators on weighted BMOA and VMOA spaces by introducing admissible weights and two pivotal test functions $\alpha$ and $\beta$ that exactly capture boundedness and compactness conditions. It demonstrates that, under broad weight hypotheses, compactness, weak compactness, complete continuity, and strict singularity are equivalent for $\psi C_{\phi}$ on $\mathrm{VMOA}_v$, with parallel (but slightly subtler) results on $\mathrm{BMOA}_v$ under additional assumptions. The paper also develops duality and density results, notably ${\rm VMOA}_v^{**}\cong {\rm BMOA}_v$, and provides explicit norm- and limit-characterizations that unify prior theorems and extend them to weighted settings. Practical criteria are given via the finite suprema of $\alpha(\psi,\phi,a)$ and $\beta(\psi,\phi,a)$ and the boundary limsup conditions, together with a suite of test functions for constructing sharp counterexamples and proving sufficiency. Overall, the results offer a robust, function-theoretic framework for analyzing weighted composition operators in a broad class of analytic function spaces with meaningful implications for operator theory in complex analysis.

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.

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

TL;DR

This work advances a comprehensive theory for weighted composition operators on weighted BMOA and VMOA spaces by introducing admissible weights and two pivotal test functions and that exactly capture boundedness and compactness conditions. It demonstrates that, under broad weight hypotheses, compactness, weak compactness, complete continuity, and strict singularity are equivalent for on , with parallel (but slightly subtler) results on under additional assumptions. The paper also develops duality and density results, notably , and provides explicit norm- and limit-characterizations that unify prior theorems and extend them to weighted settings. Practical criteria are given via the finite suprema of and and the boundary limsup conditions, together with a suite of test functions for constructing sharp counterexamples and proving sufficiency. Overall, the results offer a robust, function-theoretic framework for analyzing weighted composition operators in a broad class of analytic function spaces with meaningful implications for operator theory in complex analysis.

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 . 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.

Paper Structure

This paper contains 14 sections, 42 theorems, 297 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Consequences of John-Nirenberg's result
  5. Some general Banach space theory
  6. Some comments about the conditions concerning admissible weights
  7. Some important lemmas
  8. The functions $\alpha$ and $\beta$
  9. The function $\alpha$
  10. The function $\beta$
  11. Weighted composition operators on ${\rm BMOA} _v$ and ${\rm VMOA} _v$
  12. Compactness and related properties for $\psi C_{\phi}$ on ${\rm BMOA} _v$ and ${\rm VMOA} _v$
  13. Examples
  14. Proof of main results

Key Result

Theorem 1.1

and More specifically, for $X= {\rm BMOA} _v$ or $X= {\rm VMOA} _v$ and $\psi C_{\phi} \colon X\to X$, it holds that where the evaluation map satisfies

Theorems & Definitions (77)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Proposition 2.4
  • ...and 67 more