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

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

2009-07-15

arxiv.org/abs/0907.2489v2

Mathematics

Functional Analysis

20 pages. To appear: Indiana Univ. Math. J

Scientific paper

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.

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