Complex symmetric operators and additive preservers problem

On maps preserving the Jordan product of $C$-symmetric operators

“…for some distinct complex numbers γ u,λ and β u,λ . Hence, as Φ(λu ⊗ u) commutes with Φ(λu ⊗ u) 2 , it also commutes with the spectral projections of Φ(λu ⊗ u) 2 . Thus,…”

On maps preserving skew symmetric operators

“…In [1,Theorem 3], it was shown that a unital linear continuous map on B(H) preserves strongly skewsymmetric operators if and only if it is the identity map of B(H). We extend this result, as an immediate consequence of the previous theorem, to additive maps only.…”

“…Unfortunately, the approach used here does not allow us to obtain an analogue result of Theorem 1.1 in the setting of finite-dimensional Hilbert spaces. However, we focus to describe additive maps on B(C n ), with n ≥ 4, preserving strongly skew-symmetric operators as an extension of [1,Theorem 4].…”