Multiplicatively local spectrum-preserving maps

Let X and Y be two infinite-dimensional complex Banach spaces, and B(X) (resp. B(Y)) be the algebra of all bounded linear operators on X (resp. on Y). Fix two nonzero vectors x(0) is an element of X and y(0) is an element of Y, and let B-x0 (X)(resp. B-v0(Y)) be the collection of all operators in B(X) (resp. in B(Y)) vanishing at x(0) (resp. at y(0)). We show that if two maps phi(1) and phi(2) from B(X) onto B(Y) satisfy sigma(phi 1)(S)(phi 2),(T)(y0) = sigma(ST)(x(0)), (S, T is an element of B(X)) then cpa maps B-x0(X) onto B-y0(Y) and there exist two bijective linear mappings A : X -> Y and B : Y -> X such that Ax(0)= y(0), and phi(1)(T) = ATB for all T is an element of B(X) and phi(2)(T) = B(-1)TA(-1) for all T is not an element of B-x0(X). When X = Y = C-n, we show that the surjectivity condition on phi(1) and phi(2) is redundant. Furthermore, some known results are obtained as immediate consequences of our main results. (C) 2018 Elsevier Inc. All rights reserved.