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 | References | Similar Articles | Additional Information

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.