Abstract

We completely describe surjective maps preserving the peripheral local spectrum of Jordan product of operators. Let x0 ∈ X and y0 ∈ Y be two nonzero vectors. We show that a map ϕ from B(X) onto B(Y ) satisfies γST+T S(x0) = γϕ(S)ϕ(T )+ϕ(T )ϕ(S)(y0), (T, S ∈ B(X)), if and only if there exists a bijective bounded linear map from X into Y such thatA(x0) = y0 and either ϕ(T ) = −AT A−1 for all T ∈ B(X) or ϕ(T ) = AT A−1 for all T ∈ B(X).