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

A truncated Toeplitz operator is the compression $A_{\phi}:\K_{\Theta} \to \K_{\Theta}$ofaToeplitzoperator$T_{\phi}:H^2\to H^2$toamodelspace$\K_{\Theta} := H^2 \ominus \Theta H^2$\mathrm{.}For$\Theta$inner,let$\T_{\Theta}$denotethesetofallboundedtruncatedToeplitzoperatorson$\K_{\Theta}$. Our main result is a necessary and sufficient condition on inner functions ${\mathrm{\Theta }}_1$ and ${\mathrm{\Theta }}_2$ which guarantees that ${\mathcal{T}}_{{\mathrm{\Theta }}_1}$ and ${\mathcal{T}}_{{\mathrm{\Theta }}_2}$ are spatially isomorphic (i.e., $U\T_{\Theta_1} = \T_{\Theta_2}U$forsomeunitary$U:\K_{\Theta_1} \to \K_{\Theta_2}$). 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.