Title: Truncated Toeplitz operators: spatial isomorphism, unitary equivalence, and similarity

Authors: Joseph A. Cima, Stephan Ramon Garcia, William T. Ross and Warren R. Wogen

Issue: Volume 59 (2010), Issue 2, 595-620

Abstract: A \textit{truncated Toeplitz operator} $A_{\phi}: \mathcal{K}_{\Theta} \to \mathcal{K}_{\Theta}$ is the compression of a Toeplitz operator $T_{\phi}: H^{2} \to H^{2}$ to a model space $\mathcal{K}_{\Theta} \coloneqq H^{2} \ominus \Theta H^{2}$. For $\Theta$ inner, let $\mathcal{T}_{\Theta}$ denote the set of all bounded truncated Toeplitz operators on $\mathcal{K}_{\Theta}$. 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., $U\mathcal{T}_{\Theta_{1}} = \mathcal{T}_{\Theta_{2}}U$ for some unitary $U: \mathcal{K}_{\Theta_{1}} \to \mathcal{K}_{\Theta_{2}}$). 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.