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

Authors

William T. Ross, University of RichmondFollow
Joseph A. Cima
Stephan Ramon Garcia
Warren R. Wogen

DOI

10.1512/iumj.2010.59.4097

Abstract

A truncated Toeplitz operator Aᵩ : K_Ɵ → K_Ɵ is the compression of a Toeplitz operator Tᵩ : H² → H² to a model space K_Ɵ := H² ⊖ ^ƟH². For Ɵ inner, let T_Ɵ denote the set of all bounded truncated Toeplitz operators on K_Ɵ. Our main result is a necessary and sufficient condition on inner functions Ɵ₁ and Ɵ₂ which guarantees that T_Ɵ1 and T_Ɵ2 are spatially isomorphic. (i.e., UT_Ɵ1 = T_Ɵ2 U for some unitary U : K_Ɵ1 → K_Ɵ2). 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.

Document Type

Article

Publication Date

2010

Publisher Statement

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

DOI: 10.1512/iumj.2010.59.4097

Full Citation:

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.

Recommended Citation

Ross, William T.; Cima, Joseph A.; Garcia, Stephan Ramon; and Wogen, Warren R., "Truncated Toeplitz Operators: Spatial Isomorphism, Unitary Equivalence, and Similarity" (2010). Department of Math & Statistics Faculty Publications. 16.
https://scholarship.richmond.edu/mathcs-faculty-publications/16