Resumen de Linear maps preserving the generalized spectrum

Mostafa Mbekhta

• Let H be an infinite-dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H. For an operator T in B(H), let sg(T) denote the generalized spectrum of T. In this paper, we prove that if f: B(H) ? B(H) is a surjective linear map, then f preserves the generalized spectrum (i.e. sg(f(T)) = sg(T) for every T Î B(H)) if and only if there is A Î B(H) invertible such that either f(T) = ATA-1 for every T Î B(H), or f(T) = ATtrA-1 for every T Î B(H). Also, we prove that ?(f(T)) = ?(T) for every T Î B(H) if and only if there is U Î B(H) unitary such that either f(T) = UTU* for every T Î B(H) or f(T) = UTtrU* for every T Î B(H). Here ?(T) is the reduced minimum modulus of T.