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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Complex symmetric operators and applications II
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by Stephan Ramon Garcia and Mihai Putinar PDF
Trans. Amer. Math. Soc. 359 (2007), 3913-3931 Request permission

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 $T = CJ|T|$, where $J$ is an auxiliary conjugation commuting with $|T| = \sqrt {T^*T}$. We consider numerous examples, including the Poincaré-Neumann singular integral (bounded) operator and the Jordan model operator (compressed shift). The decomposition $T = CJ|T|$ 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
  • MR Author ID: 726101
  • 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
  • MR Author ID: 142835
  • Email: mputinar@math.ucsb.edu
  • 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
  • © Copyright 2007 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 359 (2007), 3913-3931
  • MSC (2000): Primary 30D55, 47A15
  • DOI: https://doi.org/10.1090/S0002-9947-07-04213-4
  • MathSciNet review: 2302518