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.
Categories: math.FA math.CV