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Let 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, (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. 2010 Mathematicsdoi:10.1017/s0004972711002449 fatcat:ysgxl53jh5fetgdrtfpumims3y