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

Let A 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 phi : A -> B (H): (1) phi is a *-left preserving kernel-into-range mapping, i.e., phi(T)(ker(T)) C ran(T*) for any T is an element of A. (2) phi is a generalized *-left inner derivations, i.e., phi(T) = T*A + BT for some A, B is an element of B(H). (3) phi is a generalized Jordan *-left derivations, i.e., phi(T-2) = T*phi(T) + phi(T)T - T*phi(I) T for any T c V. (4) phi is a *-left 1-preserving kernel-into-range mapping, i.e., phi(T)(ker(T)) subset of ran(T*) for any rank one operator T is an element of A. (c) 2005 Elsevier Inc. All rights reserved.