Recent progress on 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(2) (X) (resp. B-y0(2) (Y)) be the collection of all operators in B(X) (resp. in B(Y)) vanishing at x(0) (resp. at y(0)) or having zero square. On one hand, we review and discuss recent development of maps from B(X) onto B(Y) preserving the local spectrum of different products of operators and matrices. On the other hand, we establish new related results and show that if two maps phi(1) and phi(2) from B(X) onto B(Y) satisfy sigma(phi 2(T)phi 1(S)phi 2(T))(y(0)) = sigma(TST)(x(0)), (T, S is an element of B(X)), then phi(2) maps B-x0 (X) onto B-y0 (Y) and there exist two bijective linear mappings A : X -> Y and B : Y -> X, and a function alpha : B(X)\B-x0(X) -> C\{0}) such that phi(1)(T) = ATB for all T is an element of B(X), and phi(2)(T)(2) = B(-1)T(2)A(-1) and phi(2)(T)y(0) = alpha(T)B(-1)Tx(0) for all T is not an element of B-x0(2)(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.