Characterizations of left derivable maps at non-trivial idempotents on nest algebras

الأصل · EN

Let Alg N be a nest algebra associated with the nest N on a (real or complex) Banach space. Suppose that there exists a non-trivial idempotent P∈ AlgN with range P() ∈ N and δ:AlgN → AlgN is a continuous linear mapping (generalized) left derivable at P, i.e. δ(ab)=aδ(b)+bδ(a) (δ(ab)=aδ(b)+bδ(a)-baδ(I)) for any a,b∈ AlgN with ab=P. we show that δ is a (generalized) Jordan left derivation. Moreover, we characterize the strongly operator topology continuous linear maps δ on some nest algebra AlgN with property that δ(P)=2Pδ(P) or δ(P)=2Pδ(P)-Pδ(I) every idempotent P in AlgN.