Some new classes of complex symmetric operators

TL;DR

This paper explores new classes of complex symmetric operators, including binormal, algebraic degree two, and certain rank-one perturbations, providing insights into their structure and specific examples.

Contribution

It introduces new classes of complex symmetric operators and characterizes their properties, including partial isometries, expanding understanding of this operator class.

Findings

Binormal operators are complex symmetric.

Operators algebraic of degree two are complex symmetric.

Certain rank-one perturbations of normal operators are complex symmetric.

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