Let 𝓛 be a subspace lattice on a Banach space X and let δ: Alg𝓛 → B(X) be a linear mapping. If ⋁ {L ∈ 𝓛 : L₋ ⊉ L}= X or ⋁ {L₋ : L ∈ 𝓛, L₋ ⊉ L} = (0), we show that the following three conditions are equivalent: (1) δ(AB) = δ(A)B + Aδ(B) whenever AB = 0; (2) δ(AB + BA) = δ(A)B + Aδ(B) + δ(B)A + Bδ(A) whenever AB + BA = 0; (3) δ is a generalized derivation and δ(I) ∈ (Alg𝓛)'. If ⋁ {L ∈ 𝓛 : L₋ ⊉ L} = X or ⋁ {L₋ : L ∈ 𝓛, L₋ ⊉ L} = (0) and δ satisfies δ(AB + BA) = δ(A)B + Aδ(B) + δ(B)A + Bδ(A) whenever AB = 0, we show that δ is a generalized derivation and δ(I)A ∈ (Alg𝓛)' for every A ∈ Alg𝓛. We also prove that if ⋁ {L ∈ 𝓛 : L₋ ⊉ L} = X and ⋁ {L₋ : L ∈ 𝓛, L₋ ⊉ L} = (0), then δ is a local generalized derivation if and only if δ is a generalized derivation.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm206-2-2,
author = {Yunhe Chen and Jiankui Li},
title = {Mappings on some reflexive algebras characterized by action on zero products or Jordan zero products},
journal = {Studia Mathematica},
volume = {204},
year = {2011},
pages = {121-134},
zbl = {1241.47054},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm206-2-2}
}
Yunhe Chen; Jiankui Li. Mappings on some reflexive algebras characterized by action on zero products or Jordan zero products. Studia Mathematica, Tome 204 (2011) pp. 121-134. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm206-2-2/