On Generalized Derivations and Centralizers of Operator Algebras with Involution

摘要

Let B(H) be the algebra of all bounded linear operators on a complex Hilbert space H and A(H) aS dagger B(H) be a standard operator algebra which is closed under the adjoint operation. Let F: A(H)- B(H) be a linear mapping satisfying F(AA*A) = F(A)A*A + Ad(A*)A + AA*d(A) for all A a A(H), where the associated linear mapping d: A(H) - B(H) satisfies the relation d(AA*A) = d(A)A*A + Ad(A*)A + AA*d(A) for all A a A(H). Then F is of the form F(A) = SA - AT for all A a A(H) and some S, T a B(H), that is, F is a generalized derivation. We also prove some results concerning centralizers on A(H) and semisimple H (*)-algebras.