Generalized jordan derivable mappings on B(H)

Abstract

Let H be a Hilbert space over the real or complex field F and B(H) be the algebra of all bounded linear operators on H. For arbitrary fixed points C,D,M in B(H), we investigate the structure of linear mappings δ and τ on B(H) satisfying one of the following conditions: (i) δ(A)A + Aτ(A) = M for each A ∈ B(H) with A2 = I; (ii) δ(A)A + Aτ(A) = 0 for each A ∈ B(H) with A2 = 0 whenever H is infinite dimensional; (iii) δ(A)B +δ(B)A+Aτ(B)+Bτ (A) = D for all A,B ∈ B(H) 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) = TA−A(S−τ (I))−μ(A) for each A ∈ B(H), where μ is a linear mapping from B(H) into FI and T, S are fixed elements in B(H). In particular, if δ = τ , then there exist T′, S′ ∈ B(H) such that δ(A) = T′A − AS′ for each A ∈ B(H).