Truncated Toeplitz Operators: Spatial Isomorphism, Unitary Equivalence, and Similarity

TL;DR

This paper characterizes when truncated Toeplitz operator sets on different model spaces are spatially isomorphic, explores their unitary equivalence, and shows all finite-dimensional operators are similar to such operators.

Contribution

It provides a necessary and sufficient condition for spatial isomorphism of truncated Toeplitz operator sets and demonstrates their broad similarity to finite-dimensional operators.

Findings

Characterizes spatial isomorphism conditions for truncated Toeplitz operators.

Establishes criteria for unitary equivalence of these operators.

Shows every finite-dimensional operator is similar to a truncated Toeplitz operator.

Abstract

A truncated Toeplitz operator is the compression $A_{\phi}:\K_{\Theta} \to \K_{\Theta}$ of a Toeplitz operator $T_{\phi}:H^2\to H^2$ to a model space $\K_{\Theta} := H^2 \ominus \Theta H^2$. For $\Theta$ inner, let $\T_{\Theta}$ denote the set of all bounded truncated Toeplitz operators on $\K_{\Theta}$. Our main result is a necessary and sufficient condition on inner functions $\Theta_1$ and $\Theta_2$ which guarantees that $\mathcal{T}_{\Theta_1}$ and $\mathcal{T}_{\Theta_2}$ are spatially isomorphic (i.e., $U\T_{\Theta_1} = \T_{\Theta_2}U$ for some unitary $U:\K_{\Theta_1} \to \K_{\Theta_2}$). We also study operators which are unitarily equivalent to truncated Toeplitz operators and we prove that every operator on a finite dimensional Hilbert space is similar to a truncated Toeplitz operator.