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Two remarks about nilpotent operators of order two


Authors: Stephan Ramon Garcia, Bob Lutz and Dan Timotin
Journal: Proc. Amer. Math. Soc. 142 (2014), 1749-1756
MSC (2010): Primary 46Lxx, 47A05, 47B35, 47B99
Published electronically: February 19, 2014
MathSciNet review: 3168480
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Abstract: We present two novel results about Hilbert space operators which are nilpotent of order two. First, we prove that such operators are indestructible complex symmetric operators, in the sense that tensoring them with any operator yields a complex symmetric operator. In fact, we prove that this property characterizes nilpotents of order two among all nonzero bounded operators. Second, we establish that every nilpotent of order two is unitarily equivalent to a truncated Toeplitz operator.


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Additional Information

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

Bob Lutz
Affiliation: Department of Mathematics, Pomona College, Claremont, California 91711
Address at time of publication: Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, Michigan 48109-1043
Email: boblutz@umich.edu

Dan Timotin
Affiliation: Simion Stoilow Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, Bucharest 014700, Romania
Email: Dan.Timotin@imar.ro

DOI: http://dx.doi.org/10.1090/S0002-9939-2014-11944-7
Keywords: Nilpotent operator, complex symmetric operator, Toeplitz operator, model space, truncated Toeplitz operator, unitary equivalence.
Received by editor(s): June 25, 2012
Published electronically: February 19, 2014
Additional Notes: The first and second authors were partially supported by National Science Foundation Grant DMS-1001614
Communicated by: Richard Rochberg
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.