Characterization of generalized Jordan *-left derivations on real nest algebras Original Research Article

Let image be a real nest algebra of B(H), where H is a real and separable Hilbert space. We show that the following conditions are equivalent for a weak topology continuous linear map image: (1) phi is a *-left preserving kernel-into-range mapping, i.e., phi(T)(ker(T)) subset of or equal to ran(T*) for any image. (2) phi is a generalized *-left inner derivations, i.e., phi(T) = T*A + BT for some A, B set membership, variant B(H). (3) phi is a generalized Jordan *-left derivations, i.e., phi(T2) = T*phi(T) + phi(T)T − T*phi(I)T for any image. (4) phi is a *-left 1-preserving kernel-into-range mapping, i.e., phi(T)(ker(T)) subset of or equal to ran(T*) for any rank one operator image.