Complex symmetric operators and additive preservers problem

Abstract

Given a conjugation C on a separable complex Hilbert space H, a bounded linear operator T on H is said to be C-symmetric if $$CTC=T^*$$, and is said to be C-skew symmetric if $$CTC=-\,T^*$$. In this paper, we provide a complete description of all additive maps, on the algebra of all bounded linear operators acting on H, that preserve C-symmetric operators for every conjugation C. We focus also on the linear maps preserving C-skew symmetric operators.