## Pomona Faculty Publications and Research

Article

#### Department

Mathematics (Pomona)

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 $(\dim \ker T, \dim \ker T^*)$.

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