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

 

Complex symmetric operators and applications II


Authors: Stephan Ramon Garcia and Mihai Putinar
Journal: Trans. Amer. Math. Soc. 359 (2007), 3913-3931
MSC (2000): Primary 30D55, 47A15
Published electronically: March 7, 2007
MathSciNet review: 2302518
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Abstract: A bounded linear operator $ T$ on a complex Hilbert space $ \mathcal{H}$ is called complex symmetric if $ T = CT^*C$, where $ C$ is a conjugation (an isometric, antilinear involution of $ \mathcal{H}$). We prove that $ T = CJ\vert T\vert$, where $ J$ is an auxiliary conjugation commuting with $ \vert T\vert = \sqrt{T^*T}$. We consider numerous examples, including the Poincaré-Neumann singular integral (bounded) operator and the Jordan model operator (compressed shift). The decomposition $ T = CJ\vert T\vert$ also extends to the class of unbounded $ C$-selfadjoint operators, originally introduced by Glazman. In this context, it provides a method for estimating the norms of the resolvents of certain unbounded operators.


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Additional Information

Stephan Ramon Garcia
Affiliation: Department of Mathematics, University of California at Santa Barbara, Santa Barbara, California 93106-3080
Address at time of publication: Department of Mathematics, Pomona College, Claremont, California 91711
Email: garcias@math.ucsb.edu, Stephan.Garcia@pomona.edu

Mihai Putinar
Affiliation: Department of Mathematics, University of California at Santa Barbara, Santa Barbara, California 93106-3080
Email: mputinar@math.ucsb.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04213-4
PII: S 0002-9947(07)04213-4
Keywords: Complex symmetric operator, Takagi factorization, inner function, Aleksandrov-Clark operator, Clark operator, Aleksandrov measure, compressed shift, Jordan operator, $J$-selfadjoint operator, Sturm-Liouville problem.
Received by editor(s): November 9, 2004
Received by editor(s) in revised form: July 20, 2005
Published electronically: March 7, 2007
Additional Notes: This work was partially supported by the National Science Foundation Grant DMS-0350911
Article copyright: © Copyright 2007 American Mathematical Society