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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Some new classes of complex symmetric operators


Authors: Stephan Ramon Garcia and Warren R. Wogen
Journal: Trans. Amer. Math. Soc. 362 (2010), 6065-6077
MSC (2000): Primary 47B99
Published electronically: July 7, 2010
MathSciNet review: 2661508
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Abstract: We say that an operator $ T \in B(\mathcal{H})$ is complex symmetric if there exists a conjugate-linear, isometric involution $ C:\mathcal{H}\rightarrow\mathcal{H}$ so that $ T = CT^*C$. We prove that binormal operators, operators that are algebraic of degree two (including all idempotents), and large classes of rank-one perturbations of normal operators are complex symmetric. From an abstract viewpoint, these results explain why the compressed shift and Volterra integration operator are complex symmetric. Finally, we attempt to describe all complex symmetric partial isometries, obtaining the sharpest possible statement given only the data $ (\dim \ker T, \dim \ker T^*)$.


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Additional Information

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

Warren R. Wogen
Affiliation: Department of Mathematics, CB #3250, Phillips Hall, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599
Email: wrw@email.unc.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05068-8
PII: S 0002-9947(2010)05068-8
Keywords: Complex symmetric operator, normal operator, binormal operator, nilpotent operator, idempotent, partial isometry
Received by editor(s): March 17, 2009
Published electronically: July 7, 2010
Additional Notes: The first author was partially supported by National Science Foundation Grant DMS-0638789.
Article copyright: © Copyright 2010 American Mathematical Society