The Toeplitzness of weighted composition operators on Banach spaces of holomorphic functions

ABSTRACT

The concepts of various kinds of asymptotically Toeplitz operators have been originally defined and studied for the Hardy–Hilbert space. The aim of this article is to extend these definitions to the case of operators acting on Banach spaces of holomorphic functions, including Hardy spaces, Hardy–Lorentz spaces, and Hardy–Orlicz spaces. We give a function-theoretic characterization of uniformly asymptotically Toeplitzness and mean weakly asymptotically Toeplitzness for weighted composition operators. We also investigate strong and weak asymptotic Toeplitzness of weighted composition operators.

KEYWORDS:

• Weighted composition operators
• asymptotically Toeplitz operators
• hardy spaces

AMS SUBJECT CLASSIFICATIONS:

• 47B33
• 47B35

Acknowledgements

We would like to thank the referee for pointing out the strength of the conjectured characterization of mean weakly asymptotically Toeplitzness for a weighted composition operator.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The second author is supported in part by Engineering and Physical Sciences Research Council grant EP/X024555/1. The third author is supported by the National Science Center (Narodowe Centrum Nauki), Poland [project number 2017/26/D/ST1/00060]. The fourth author is grateful for the financial support from the Doctoral Network in Information Technologies and Mathematics at Åbo Akademi University.