On the Norm of Jordan $*$-Derivations

Document Type : Original Article


Department of Mathematics, Faculty of basic sciences, University of Bojnord, P.O. Box 1339, Bojnord, Iran.


Let $\mathcal H$ be a complex Hilbert space and let $B(\mathcal H)$ be the algebra of all bounded linear operators on $\mathcal H$. Let $T\in\ B(\mathcal H)$.
In this paper, we determine the norm of the inner Jordan $*$-derivation $\Delta_T:X\mapsto TX-X^*T$ acting on the Banach algebra $B(\mathcal{H})$. More precisely, we show that $$\big{\|}\Delta_T\big{\|}\geq 2\sup_{\lambda\in W_0(T)}|{\rm Im}(\lambda)|$$
in which $W_0(T)$ is the maximal numerical range of operator $T$.