## 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.## References

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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