Characterizing isomorphisms between standard operator algebras by spectral functions

Let A and B be standard operator algebras on an infinite dimensional complex Banach space X, and let Phi be a map from A onto B. We introduce thirteen parts of spectrum for elements in A and 8 and let Delta(A)(T) denote any one of these thirteen parts of the spectrum of T in A. We show that if Phi satisfies that Delta(A)(T + S) = Delta(B)(Phi(T) + Phi(S)) and Delta(A)(T + 2S) = Delta(B)(Phi(T) + 2 Phi(S)) for all T, S is an element of A, then Phi is either an isomorphism or an anti-isomorphism.