Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with the Center of Excellence in Analysis on Algebraic Structures and Tusi Mathematical Research Group)Khayyam Journal of Mathematics2423-47886120200101On the Norm of Jordan $*$-Derivations1041079717610.22034/kjm.2019.97176ENAbolfazlNiazi MotlaghDepartment of Mathematics, Faculty of basic sciences, University of Bojnord,
P.O. Box 1339, Bojnord, Iran.Journal Article20190121Let $\mathcal H$ be a complex Hilbert space and let $B(\mathcal H)$ be the algebra of all bounded linear operators on $\mathcal H$. Let $T\in\ B(\mathcal H)$.<br />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 $$\big{\|}\Delta_T\big{\|}\geq 2\sup_{\lambda\in W_0(T)}|{\rm Im}(\lambda)|$$<br />in which $W_0(T)$ is the maximal numerical range of operator $T$.http://www.kjm-math.org/article_97176_b3a4d842feb70c384bb0001e22fc2e2b.pdf