Some new classes of complex symmetric operators
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- by Stephan Ramon Garcia and Warren R. Wogen PDF
- Trans. Amer. Math. Soc. 362 (2010), 6065-6077 Request permission
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 $(\dim \ker T, \dim \ker T^*)$.References
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Additional Information
- Stephan Ramon Garcia
- Affiliation: Department of Mathematics, Pomona College, Claremont, California 91711
- MR Author ID: 726101
- 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
- MR Author ID: 183945
- Email: wrw@email.unc.edu
- 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.
- © Copyright 2010 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 362 (2010), 6065-6077
- MSC (2000): Primary 47B99
- DOI: https://doi.org/10.1090/S0002-9947-2010-05068-8
- MathSciNet review: 2661508