Truncated Toeplitz operators: spatial isomorphism, unitary equivalence, and similarity

Abstract

A truncated Toeplitz operator A ϕ : K Θ → K Θ is the compression of a Toeplitz operator T ϕ : H 2 → H 2 to a model space K Θ := H 2 e ΘH 2 . For Θ 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 Θ 1 and Θ 2 which guarantees that T Θ1 and TΘ 2 are spatially isomorphic (i.e., UT Θ1 = T Θ2 U for some unitary U: K Θ1 → K Θ2 ). 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.