Maps preserving the local spectrum of skew-product of operators

Abstract

Let H and K be infinite dimensional complex Hilbert spaces and let B(H) be the algebra of all bounded linear operators on H. Let σT(h) denote the local spectrum of an operator T∈B(H) at any vector h∈H, and fix two nonzero vectors h0∈H and k0∈K. We show that if a map φ:B(H)→B(K) has a range containing all operators of rank at most two and satisfiesσTS⁎(h0)=σφ(T)φ(S)⁎(k0) for all T, S∈B(H), then there exist two unitary operators U and V in B(H,K) such that Uh0=αk0 for some nonzero α∈C and φ(T)=UTV⁎ for all T∈B(H). We also described maps φ:B(H)→B(K) satisfyingσTS⁎T(h0)=σφ(T)φ(S)⁎φ(T)(k0) for all T, S∈B(H), and with the range containing all operators of rank at most four.