A truncated Toeplitz operator Aᵩ : KƟ → KƟ is the compression of a Toeplitz operator Tᵩ : H2 → H2 to a model space KƟ := H2 ⊖ ƟH2. For Ɵ inner, let TƟ denote the set of all bounded truncated Toeplitz operators on KƟ. Our main result is a necessary and sufficient condition on inner functions Ɵ1 and Ɵ2 which guarantees that TƟ1 and TƟ2 are spatially isomorphic. (i.e., UTƟ1 = TƟ2 U for some unitary U : KƟ1 → KƟ2). 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.
Copyright © 2010 Indiana University Mathematics Journal. This article first appeared in Indiana University Mathematics Journal 59:2 (2010), 595-620.
The definitive version is available at: http://www.iumj.indiana.edu/IUMJ/FULLTEXT/2010/59/4097
Cima, Joseph A., Stephan Ramon Garcia, William T. Ross, and Warren R. Wogen. "Truncated Toeplitz Operators: Spatial Isomorphism, Unitary Equivalence, and Similarity." Indiana University Mathematics Journal 59, no. 2 (2010): 595-620. doi:10.1512/iumj.2010.59.4097.
Ross, William T.; Cima, Joseph A.; Garcia, Stephan Ramon; and Wogen, Warren R., "Truncated Toeplitz Operators: Spatial Isomorphism, Unitary Equivalence, and Similarity" (2010). Math and Computer Science Faculty Publications. 16.