Similarity orbits of complex symmetric operators

Abstract

An operator $T$ on a complex Hilbert space $\mathcal{H}$ is said to be complex symmetric if $T$ can be represented as a symmetric matrix relative to some orthonormal basis for $\mathcal{H}$. In this article we explore the stability of complex symmetry under the condition of similarity. It is proved that the similarity orbit of an operator $T$ is included in the class of complex symmetric operators if and only if $T$ is an algebraic operator of degree at most $2$.