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 x0∈X and y0∈Y, and let Bx0(X) (resp. By0(Y)) be the collection of all operators in B(X) (resp. in B(Y)) vanishing at x0 (resp. at y0). We show that if two maps φ1 and φ2 from B(X) onto B(Y) satisfyσφ1(S)φ2(T)(y0)=σST(x0),(S,T∈B(X)), then φ2 maps Bx0(X) onto By0(Y) and there exist two bijective linear mappings A:X→Y and B:Y→X such that Ax0=y0, and φ1(T)=ATB for all T∈B(X) and φ2(T)=B−1TA−1 for all T∉Bx0(X). When X=Y=Cn, we show that the surjectivity condition on φ1 and φ2 is redundant. Furthermore, some known results are obtained as immediate consequences of our main results.