No abstract

No 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.

Let M n (C) denote the algebra of all n × n complex matrices, and x 0 a nonzero vector in C n . For two fixed scalars μ and ν in C for which (μ,ν) = (0,0) , we characterize all maps ϕ on M n (C) satisfyingThis provides, in particular, a complete description of all maps on M n (C) preserving the local spectrum of the skew double product " T S * " or the skew triple product " T S * T " of matrices. It also unifies and extends several known results on local spectrum preservers. (2010): Primary 47B49, Secondary 47A10, 47A11. Mathematics subject classification

Abstract. Let H and K be two infinite-dimensional complex Hilbert spaces, and fix two nonzero vectors h 0 ∈ H and k 0 ∈ K . Let L (H ) (resp. L (K ) ) denote the algebra of all bounded linear operators on H (resp. on K ), and letif and only if there exist a unitary operator U from H into K and a scalar α ∈ C such that Uh 0 = αk 0 and ϕ(T ) = λUTU * for all T ∈ L (H ) , where λ is a scalar of modulus 1 .Mathematics subject classification (2010): Primary 47B49, Secondary 47A10, 47A11.