On linear maps preserving complex symmetry

Abstract

An operator T on a complex Hilbert space H is said to be complex symmetric if there exists a conjugate-linear, isometric involution C:H→H so that CTC=T⁎. This paper is devoted to describing which linear maps leave the class of complex symmetric operators invariant. Complete characterizations are obtained for several classes of linear maps, including similarity transformations, surjective linear isometries, multiplication operators and certain completely positive maps.