CHARACTERIZATIONS OF JORDAN DERIVATIONS ON STRONGLY DOUBLE TRIANGLE SUBSPACE LATTICE ALGEBRAS

Abstract

AbstractLet 𝒟 be a strongly double triangle subspace lattice on a nonzero complex reflexive Banach space X and let δ:Alg 𝒟→Alg 𝒟 be a linear mapping. We show that δ is Jordan derivable at zero, that is, δ(AB+BA)=δ(A)B+Aδ(B)+δ(B)A+Bδ(A) whenever AB+BA=0 if and only if δ has the form δ(A)=τ(A)+λA for some derivation τ and some scalar λ. We also show that if the dimension of X is greater than 2, then δ satisfies δ(AB+BA)=δ(A)B+Aδ(B)+δ(B)A+Bδ(A) whenever AB=0 if and only if δ is a derivation.