0:00 Lie Algebra. Sets Derivation And The Orthogonal Projection At Lie Derivation Characteristics

Some linear mappings on operators algebras, such as isomorphism, derivations, Lie derivations, people has been studying. Let H be a Hilbert space, N be a completed nest of closed subspace of H, and Alg N be the associated nest algebra, ifδ:Alg N→Alg N is a linear map satisfyingδ([a,b])= [δ(a),b]+[a,δ(b)] for all a,b∈Alg N with ab = 0. Then there existsr∈Alg N and a linear mapτ:Alg N→CI vanishing at commuta-tors[a,b]when ab= 0such thatδ(a)=ra-ar+τ(a)I for any a∈Alg N.Let H be a Hilbert space, N be a nontrivial nest of closed subspace of H, and Alg N be the associated nest algebra. Let M be nontrivial subspace in N, and p be the orthogonal projection on M. Ifδ:Alg N→Alg N is a linear map satisfyingδ([a,b]) = [δ(a),b]+[a,δ(b)]for all a,b∈Alg N with ab= p,thenδ= d+τ, where d is a derivation of Alg N andτ:Alg N→CI is a linear map vanishing at commutators [a,b] with ab=p.