Research Article

Generalized Jordan derivable mappings on B ( H )

Published in: Quaestiones Mathematicae
Volume 48 , issue 7 : Entrepreneurship and Africa’s Cultural Context , 2024 , pages: 993–1005
DOI: 10.2989/16073606.2025.2459365
Author(s): Kaijia Luo Institute of Mathematics, Hangzhou Dianzi University, China, Jiankui Li School of Mathematics, East China University of Science and Technology, China, Shanshan Su School of Mathematics, East China University of Science and Technology, China,
Keywords: 47B47, 47L30, 39B42, Derivation, generalized Jordan derivable mapping, functional identity,
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Abstract

Let be a Hilbert space over the real or complex field and be the algebra of all bounded linear operators on . For arbitrary fixed points C, D, M in , we investigate the structure of linear mappings δ and τ on satisfying one of the following conditions: (i) δ(A)A + Aτ(A) = M for each with A 2 = I ; (ii) δ(A)A + Aτ(A) = 0 for each with A 2 = 0 whenever is infinite dimensional; (iii) δ(A)B + δ(B)A + Aτ (B) + Bτ(A) = D for all with AB +BA = C. In every case δ, τ are of the form δ(A) = (S +δ(I))A – AT +µ(A) and τ(A) = T A − A(S − τ(I )) − µ(A) for each , where µ is a linear mapping from into and T, S are fixed elements in . In particular, if δ = τ , then there exist such that δ(A) = T′ A – AS′ for each .
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