Product of Matrix Valued Truncated Toeplitz Operators

TL;DR

This paper investigates conditions under which the product of two matrix valued truncated Toeplitz operators remains within the same class, extending understanding of their algebraic structure in vector-valued function spaces.

Contribution

It provides necessary and sufficient conditions for the product of two matrix valued truncated Toeplitz operators to also be a truncated Toeplitz operator, under certain assumptions.

Findings

Characterization of when the product $A_\Phi A_\Psi$ is a truncated Toeplitz operator.

Extension of scalar results to matrix valued operators.

Conditions depend on properties of the inner function and symbols.

Abstract

Let $A_\Phi$ be a matrix valued truncated Toeplitz operator-the compression of multiplication operator to vector-valued model space $H^2(E)\ominus \Theta H^2(E)$, where $\Theta$ is a matrix valued non constant inner function. Under supplementary assumptions, we find necessary and sufficient condition that the product $A_\Phi A_\Psi$ is itself a matrix valued truncated Toeplitz operator.