Characterizations of additive local Jordan *-derivations by action at idempotents

Abstract

Let H be a real or complex Hilbert space and B ( H ) the algebra of all bounded linear operators on H. Recall that a map δ : B ( H ) → B ( H ) is called an inner Jordan ∗ -derivation if there exists some T ∈ B ( H ) such that δ ( A ) = AT − T A ∗ for all A ∈ B ( H ) . In this paper, it is proved that inner Jordan ∗ -derivations are the only additive maps δ of B ( H ) with the property that δ ( P ) = δ ( P ) P ∗ + Pδ ( P ) for all idempotent operators P ∈ B ( H ) if dim ⁡ H = ∞ , which is satisfied by additive local Jordan ∗ -derivations. For the finite dimensional case, additional conditions are required for δ to be an inner Jordan ∗ -derivation. As applications, it is shown that, for any given C , D ∈ B ( H ) , δ satisfies δ ( A ) B ∗ + Bδ ( A ) + δ ( B ) A ∗ + Aδ ( B ) = D for all A , B ∈ B ( H ) with AB + BA = C if and only if δ is an inner Jordan ∗ -derivation and D = δ ( C ) . Also, several known results are generalized.