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

A truncated Toeplitz operator A phi : K(Theta) -> K(Theta) is the compression of a Toeplitz operator T(phi): H(2) -> H(2) to a model space K(Theta) := H(2) circle minus 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 T(Theta 1) and T(Theta 2) are spatially isomorphic (i.e., UT(Theta 1), = T(Theta 2)U for some unitary U : K(Theta 1) -> 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.