Refinements of numerical radius inequalities via Specht’s rati‏o

Yaser Khatib, Mahmoud Hassani, Maryam Amyari

Abstract


‎We present some new numerical radius inequalities of Hilbert space‎

‎operators‎.

‎We improve and generalize some inequalities with respect to Specht's ratio‎. ‎Let $A$ and $B$ be two positive invertible operators on a Hilbert space $H$ and let $X$ be a bounded operator on $H$‎. ‎Then‎

‎\begin{equation*}‎

‎\omega((A\natural B)X)\leq \frac{1}{2S(\sqrt{h})}\|X^*BX+A\|,\quad (h>0,\‎, ‎h\neq 1)‎

‎\end{equation*}‎

‎where $\|\cdot\|,\,\,\,\omega(\cdot),\,\,\‎, ‎S(\cdot),$ and $\natural$ denote the usual operator norm‎, ‎numerical radius‎, ‎the Specht's ratio‎, ‎and the operator geometric mean‎, ‎respectively‎.


Keywords


positive operators‎, ‎normalized positive linear map‎, ‎numerical radius‎, ‎Specht's ratio.‎

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