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 σ T (h) denote the local spectrum of an operator T ∈ B(H ) at any vector h ∈ H , and fix two nonzero vectors h 0 ∈ H and k 0 ∈ K . We show that if a map ϕ : B(H ) → B(K ) has a range containing all operators of rank at most two and satisfiesfor all T, S ∈ B(H ), then there exist two unitary operators U and V in B(H , K ) such that Uh 0 = αk 0 for some nonzero α ∈ C and ϕ(T ) = UT V * for all T ∈ B(H ). We also described maps ϕ :for all T, S ∈ B(H ), and with the range containing all operators of rank at most four.