$C^*$-algebras generated by truncated Toeplitz operators

TL;DR

This paper extends Coburn's description of Toeplitz algebras to truncated Toeplitz operators, providing new insights into their structure and examples of complex symmetric operators not equivalent to those with continuous symbols.

Contribution

It offers an analogue of Coburn's theorem for truncated Toeplitz operators and presents examples of complex symmetric operators outside the class of those with continuous symbols.

Findings

Established an analogue of Coburn's description for truncated Toeplitz algebras

Provided examples of complex symmetric operators not unitarily equivalent to truncated Toeplitz operators with continuous symbols

Enhanced understanding of the structure and diversity of truncated Toeplitz operators

Abstract

We obtain an analogue of Coburn's description of the Toeplitz algebra in the setting of truncated Toeplitz operators. As a byproduct, we provide several examples of complex symmetric operators which are not unitarily equivalent to truncated Toeplitz operators having continuous symbols.