Remarks on the Structure of Complex Symmetric Operators

Abstract.

A conjugation C is antilinear isometric involution on a complex Hilbert space ${\mathcal{H}}$ , and $T \in {\mathcal{B}}({\mathcal{H}})$ is called complex symmetric if T* = CTC for some conjugation C. We use multiplicity theory to describe all conjugations commuting with a fixed positive operator. We expand upon a result of Garcia and Putinar to provide a factorization of complex symmetric operators which is based on the polar decomposition.