Document Type: Research Paper

Authors

Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran

Abstract

We obtain some inequalities related to the powers of numerical radius inequalities of Hilbert space operators. Some results that employ the Hermite-Hadamard inequality for vectors in normed linear spaces are also obtained. We improve and generalize some inequalities with respect to Specht's ratio. Among them, we show that, if $A, B\in \mathcal{B(\mathcal{H})}$ satisfy in some conditions, it follows that
\begin{equation*}
\omega^2(A^*B)\leq
\frac{1}{2S(\sqrt{h})}\Big\||A|^{4}+|B|^{4}\Big\|-\displaystyle{\inf_{\|x\|=1}}
\frac{1}{4S(\sqrt{h})}\big(\big\langle \big(A^*A-B^*B\big)
x,x\big\rangle\big)^2
\end{equation*}
for some $h>0$, where $\|\cdot\|,\,\,\,\omega(\cdot)$ and $S(\cdot)$ denote the usual operator norm, numerical radius and the Specht's ratio, respectively.

Keywords

Main Subjects

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