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

Let H be a complex separable Hilbert space, with dim H >= 4, and let B(H) be the algebra of all bounded linear operators acting on H. Given a conjugate-linear isometric involution C : H -> H, an operator T. B(H) is called C-skew symmetric if it satisfies CTC = -T*. In the present paper, we characterize all those maps Phi : B(H) -> B(H) that satisfy the following: TS is C-skew symmetric double left right arrow Phi(T)Phi(S) is C-skew symmetric for every conjugate-linear isometric involution C on H and all T, S is an element of B(H).