Some Inequalities for the Numerical Radius and Rhombic Numerical Radius

Akram Babri Bajmaeh, Mohsen Erfanian Omidvar

In this paper, the definition Rhombic numerical radius is introduced and we present several numerical radius inequalities. Some applications of these inequalities are considered as well. Particular, it is shown that, if $A\in \mathcal B \left( \mathcal{H} \right)$ with the Cartesian decomposition $A=C+iD$ and $r\geq 1$, then \[\begin{aligned} mega^r(A)eq\frac{qrt{2}}{2}{ eftVertvert C+D\rvert^{2 r}+vert C-D\rvert^{2 r}\right\rVert}^{\frac{1}{2}}. \end{aligned}\]