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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 alsostudy operators which are unitarily equivalent to truncated Toeplitz operatorsand we prove that every operator on a finite dimensional Hilbert space issimilar to a truncated Toeplitz operator.

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