Multipliers and equivalence of functions, spaces, and operators

TL;DR

This paper develops a unified framework for understanding the equivalence of various generalized Toeplitz operators on L^2 using multipliers, extending classical results and covering multiple operator classes.

Contribution

It introduces a comprehensive approach to operator equivalence via multipliers, generalizing existing theorems to a broader class of Toeplitz-related operators.

Findings

Unified criteria for operator equivalence

Extension of Brown–Halmos theorem

Application to multiple operator classes

Abstract

This paper offers a unified approach to determining when two generalized Toeplitz operators on L^2 are equivalent. This will be done through multipliers between closed subspaces of L^2. Our discussion will include Toeplitz operators (and their duals) on the Hardy space, Hankel operators, asymmetric truncated Toeplitz operators, and dual asymmetric truncated Toeplitz operators. Along the way, there will be a discussion of equivalence of functions and kernels of generalized Toeplitz operators and a generalization of the Brown--Halmos theorem for this class of operators.