On the Norm of Jordan $*$-Derivations

Abstract

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 $Tin B(mathcal H)$.<br />In this paper, we determine the norm of the inner Jordan $*$-derivation $Delta_T:Xmapsto TX-X^*T$ acting on the Banach algebra $B(mathcal{H})$. More precisely, we show that $$big{|}Delta_Tbig{|}geq 2sup_{lambdain W_0(T)}|{rm Im}(lambda)|$$<br />in which $W_0(T)$ is the maximal numerical range of operator $T$.