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

Abstract

A truncated Toeplitz operator is the compression $A_ϕ:\K_Θ \to \K_Θ$ of a Toeplitz operator $T_{\varphi }:H^2\to H^2$ to a model space $\K_Θ := H^2 \ominus ΘH^2$. For $\Theta$ inner, let $\T_Θ$ denote the set of all bounded truncated Toeplitz operators on $\K_Θ$. 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., $U\T_{Θ_1} = \T_{Θ_2}U$ for some unitary $U:\K_{Θ_1} \to \K_{Θ_2}$). 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.

20 pages. To appear: Indiana Univ. Math. J