Jordan derivable mappings on \(B(H)\)

Abstract

Let \(H\) be a real or complex Hilbert space with the dimension greater than one and \(B(H)\) the algebra of all bounded linear operators on \(H\). Assume that \(\delta\) is a linear mapping from \(B(H)\) into itself which is Jordan derivable at a given element \(\Omega\in B(H)\), in the sense that \(\delta(A\circ B)=\delta(A)\circ B+A\circ\delta (B)\) holds for all \(A,B\in B(H)\) with \(A\circ B = \Omega\), where \(\circ\) denotes the Jordan product \( {A\circ B } =AB+BA\). In this paper, we show that if \(\Omega\) is an arbitrary but fixed nonzero operator, then \(\delta\) is a derivation; if \(\Omega\) is a zero operator, then \(\delta\) is a generalized derivation.