Factorization of Toeplitz operators

TL;DR

This paper investigates the factorization of positive Toeplitz operators associated with commuting contractions on Hilbert spaces, revealing new factorization results and differences between cases with two and more than two operators.

Contribution

It introduces novel factorization theorems for positive T-Toeplitz operators using isometric pseudo-extensions, including for pure lower T-Toeplitz operators, and explores the case of (S,T)-Toeplitz operators.

Findings

Positive T-Toeplitz operators can be factorized via isometric pseudo-extensions.

A distinct factorization exists for positive pure lower T-Toeplitz operators.

Differences are observed between the cases d=2 and d>2 in the factorization process.

Abstract

In this article, by considering $T=(T_1,\dots, T_d)$, an $d$-tuple of commuting contractions on a Hilbert space $\mathcal{H}$, we study $T$-Toeplitz operators which consists of bounded operators $X$ on $\mathcal{H}$ such that \[ T_i^*XT_i=X \] for all $i=1,\dots,d$. We show that any positive $T$-Toeplitz operator can be factorized in terms of an isometric pseudo-extension of $T$. A similar factorization result is also obtained for positive pure lower $T$-Toeplitz operators. However, the latter factorization is obtained in terms of a special type of isometric pseudo-extension of $T$, and a certain difference has been observed between the case $d=2$ and $d>2$. In a more general context, by considering $d$-tuples of commuting contractions $S$ and $T$, we also study $(S, T)$-Toeplitz operators.