Maps preserving peripheral spectrum of Jordan semi-triple products of operators

摘要

Let A and B be (not necessarily unital or closed) standard operator algebras on complex Banach spaces X and Y, respectively. For a bounded linear operator A on X, the peripheral spectrum σπ(A) of A is the set σπ (A)={z∈σ(A):|z|=maxω∈ σ(A)|ω|}, where σ(A) denotes the spectrum of A. Assume that Φ:A→B is a map the range of which contains all operators of rank at most two. It is shown that the map Φ satisfies the condition that σπ (BAB)= σπ (Φ(B)Φ(A)Φ(B)) for all A,B∈A if and only if there exists a scalar λ∈? with λ3=1 and either there exists an invertible operator T∈B(X*,Y) such that Φ(A)=λTA *T-1 for every A∈A; or there exists an invertible operator T∈B(X,Y) such that Φ(A)=λ TA *T-1 for every A∈A. If X=H and Y=K are complex Hilbert spaces, the maps preserving the peripheral spectrum of the Jordan skew semi-triple product BA*B are also characterized. Such maps are of the form AUAU or A→UAtU, where U∈B(H,K) is a unitary operator, At denotes the transpose of A in an arbitrary but fixed orthonormal basis of H.