Maps preserving the local spectrum of skew-product of operators

Let HH and KK be infinite dimensional complex Hilbert spaces and let B(H)B(H) be the algebra of all bounded linear operators on HH. Let σT(h)σT(h) denote the local spectrum of an operator T∈B(H)T∈B(H) at any vector h∈Hh∈H, and fix two nonzero vectors h0∈Hh0∈H and k0∈Kk0∈K. We show that if a map φ:B(H)→B(K)φ:B(H)→B(K) has a range containing all operators of rank at most two and satisfiesσTS⁎(h0)=σφ(T)φ(S)⁎(k0)σTS⁎(h0)=σφ(T)φ(S)⁎(k0) for all T, S∈B(H)T, S∈B(H), then there exist two unitary operators U and V   in B(H,K)B(H,K) such that Uh0=αk0Uh0=αk0 for some nonzero α∈Cα∈C and φ(T)=UTV⁎φ(T)=UTV⁎ for all T∈B(H)T∈B(H). We also described maps φ:B(H)→B(K)φ:B(H)→B(K) satisfyingσTS⁎T(h0)=σφ(T)φ(S)⁎φ(T)(k0)σTS⁎T(h0)=σφ(T)φ(S)⁎φ(T)(k0) for all T, S∈B(H)T, S∈B(H), and with the range containing all operators of rank at most four.