Let 𝓛 be a 𝒥-subspace lattice on a Banach space X and Alg 𝓛 the associated 𝒥-subspace lattice algebra. Assume that δ: Alg 𝓛 → Alg 𝓛 is an additive map. It is shown that δ satisfies δ(AB + BA) = δ(A)B + Aδ(B) + δ(B)A + Bδ(A) for any A,B ∈ Alg 𝓛 with AB + BA = 0 if and only if δ(A) = τ(A) + δ(I)A for all A, where τ is an additive derivation; if X is complex with dim X ≥ 3 and if δ is linear, then δ satisfies δ(AB + BA) = δ(A)B + Aδ(B) + δ(B)A + Bδ(A) for any A,B ∈ Alg 𝓛 with AB + BA = I if and only if δ is a derivation.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm210-1-2,
author = {Xiaofei Qi},
title = {Characterization of Jordan derivations on J-subspace lattice algebras},
journal = {Studia Mathematica},
volume = {209},
year = {2012},
pages = {17-33},
zbl = {1250.47084},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm210-1-2}
}
Xiaofei Qi. Characterization of Jordan derivations on 𝒥-subspace lattice algebras. Studia Mathematica, Tome 209 (2012) pp. 17-33. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm210-1-2/