Abstract
Let M-n(C) be the algebra of all n x n complex matrices, and fix a nonzero vector x(0) is an element of C-n. For any matrix T is an element of M-n(C), let sigma(T) be its spectrum and sigma(T) (x(0)) be its local spectrum at x(0). We show that a map phi on M-n(C) satisfies
sigma phi(T)phi(S)-phi(S)phi(T)* (x0) = sigma TS-ST * (x(0)), (T, S is an element of M-n(C)) if and only if there exists a unitary matrix U. M n(C) and a nonzero scalar a such that U-x0 = ax(0) and (T) = +/- UTU * for all T is an element of M-n(C). To prove this result, we also describe the form of all maps. on M-n(C) satisfying
sigma (phi(T)phi(S) -phi(S)phi(T)*) = sigma (TS - ST *), (T, S is an element of M-n(C)).
As immediate consequences, we characterize all maps on M-n(C) preserving the local spectrum and spectrum of the skew Jordan product of matrices.