Characterizations of complex symmetric Toeplitz operators

TL;DR

This paper provides a complete characterization of complex symmetric Toeplitz operators, linking them to $S$-Toeplitz operators and transpose properties, and addresses a longstanding open question in the field.

Contribution

It introduces a new characterization of complex symmetric Toeplitz operators via $S$-Toeplitz conditions and canonical conjugation factorizations, solving a major open problem.

Findings

Every conjugation admits a canonical factorization.

A Toeplitz operator is complex symmetric iff it is $S$-Toeplitz for some unilateral shift $S$.

Characterization of complex symmetric Toeplitz operators on the Hardy space over the polydisc.

Abstract

We present complete characterizations of Toeplitz operators that are complex symmetric. This follows as a by-product of characterizations of conjugations on Hilbert spaces. Notably, we prove that every conjugation admits a canonical factorization. As a consequence, we prove that a Toeplitz operator is complex symmetric if and only if the Toeplitz operator is $S$-Toeplitz for some unilateral shift $S$ and the transpose of the Toeplitz operator matrix is equal to the matrix of the Toeplitz operator corresponding to the basis of the unilateral shift $S$. Also, we characterize complex symmetric Toeplitz operators on the Hardy space over the open unit polydisc. Our results answer the well known open question about characterizations of complex symmetric Toeplitz operators.