Refinements of numerical radius inequalities via Specht’s ratio
Keywords:
positive operators, normalized positive linear map, numerical radius, Specht's ratio.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.
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