Characterization of truncated Toeplitz operators by conjugations

TL;DR

This paper explores how conjugations can be used to characterize truncated Toeplitz operators, providing new insights into their structure and a characterization of classical Toeplitz operators.

Contribution

It demonstrates that C-symmetry with respect to a family of conjugations characterizes truncated Toeplitz operators in certain model spaces.

Findings

C-symmetry with multiple conjugations characterizes truncated Toeplitz operators

A new characterization of classical Toeplitz operators via conjugations

Results hold for specific classes of inner functions and model spaces

Abstract

Truncated Toeplitz operators are C--symmetric with respect to the canonical conjugation given on an appropriate model space. However, by considering only one conjugation one cannot characterize truncated Toeplitz operators. It will be proved, for some classes of inner functions and the model spaces connected with them, that if an operator on a model space is C--symmetric for a certain family of conjugations in the model space, then is has to be truncated Toeplitz. A characterization of classical Toeplitz operators is also presented in terms of conjugations.