Non-linear Preservers of the Product of C-Skew Symmetry

Abstract

Let H be a complex separable Hilbert space, with $\dim H \ge 4$, and let ${{\mathcal {B}}}(H)$ be the algebra of all bounded linear operators acting on H. Given a conjugate-linear isometric involution $C:H\rightarrow H$, an operator $T\in {{\mathcal {B}}}(H)$ is called C-skew symmetric if it satisfies $CTC=-T^*$. In the present paper, we characterize all those maps $\Phi :{{\mathcal {B}}}(H)\rightarrow {{\mathcal {B}}}(H)$ that satisfy the following:

$$\begin{aligned} TS \text{ is } C\text{-skew } \text{ symmetric } \;\;\Longleftrightarrow \;\;\Phi (T)\Phi (S) \text{ is } C\text{-skew } \text{ symmetric } \end{aligned}$$

for every conjugate-linear isometric involution C on H and all $T,S \in {{\mathcal {B}}}(H)$.