Document Type

Article

Department

Mathematics (Pomona)

Publication Date

2010

Keywords

Complex symmetric operator, normal operator, binormal operator, nilpotent operator, idempotent, partial isometry

Abstract

We say that an operator $T \in B(H)$ is complex symmetric if there exists a conjugate-linear, isometric involution $C:H\to H$ 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 $(\dim \ker T, \dim \ker T^*)$.

Comments

First published in Transactions of the American Mathematical Society in vol. 362 no. 11, 2010, published by the American Mathematical Society.

Rights Information

© 2010 American Mathematical Society

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