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Unitary equivalence to a truncated Toeplitz operator: analytic symbols


Authors: Stephan Ramon Garcia, Daniel E. Poore and William T. Ross
Journal: Proc. Amer. Math. Soc. 140 (2012), 1281-1295
MSC (2010): Primary 47A05, 47B35, 47B99
DOI: https://doi.org/10.1090/S0002-9939-2011-11060-8
Published electronically: July 22, 2011
MathSciNet review: 2869112
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Abstract: Unlike Toeplitz operators on $ H^2$, truncated Toeplitz operators do not have a natural matricial characterization. Consequently, these operators are difficult to study numerically. In this paper we provide criteria for a matrix with distinct eigenvalues to be unitarily equivalent to a truncated Toeplitz operator having an analytic symbol. This test is constructive, and we illustrate it with several examples. As a byproduct, we also prove that every complex symmetric operator on a Hilbert space of dimension $ \leq 3$ is unitarily equivalent to a direct sum of truncated Toeplitz operators.


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Additional Information

Stephan Ramon Garcia
Affiliation: Department of Mathematics, Pomona College, Claremont, California 91711
Email: Stephan.Garcia@pomona.edu

Daniel E. Poore
Affiliation: Department of Mathematics, Pomona College, Claremont, California 91711
Email: dep02007@mymail.pomona.edu

William T. Ross
Affiliation: Department of Mathematics and Computer Science, University of Richmond, Richmond, Virginia 23173
Email: wross@richmond.edu

DOI: https://doi.org/10.1090/S0002-9939-2011-11060-8
Keywords: Toeplitz operator, model space, truncated Toeplitz operator, reproducing kernel, complex symmetric operator, conjugation, hyperbolic geometry, Euclid, Hilbert’s axioms, pseudo-hyperbolic metric, hyperbolic metric, Poincaré model, trace.
Received by editor(s): December 21, 2010
Published electronically: July 22, 2011
Additional Notes: The first author was partially supported by National Science Foundation Grant DMS-1001614.
Communicated by: Richard Rochberg
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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