Some refinements of numerical radius inequalities

Z.  Heydarbeygi; M.  Amyari; M.  Khanehgir

doi:10.37863/umzh.v72i10.6027

Z. Heydarbeygi Mashhad Branch, Islamic Azad Univ., Iran
M. Amyari Mashhad Branch, Islamic Azad Univ., Iran
M. Khanehgir Mashhad Branch, Islamic Azad Univ., Iran

DOI: https://doi.org/10.37863/umzh.v72i10.6027

Abstract

UDC 517.5

In this paper, we give some refinements for the second inequality in $\dfrac{1}{2}\|A\| \leq w(A) \leq \|A\|,$ where $A\in B(H).$ In particular, if $A$ is hyponormal by refining the Young inequality with the Kantorovich constant $K(\cdot, \cdot),$ we show that $w(A)\leq \dfrac{1}{\displaystyle {2\inf\nolimits_{\| x \|=1}}\zeta(x)}\| |A|+|A^{*}|\|\leq \dfrac{1}{2}\| |A|+|A^*|\|,$ where $\zeta(x)=K\left(\dfrac{\langle |A|x,x \rangle}{\langle |A^{*}|x,x \rangle},2\right)^{r},$ $r=\min\{\lambda,1-\lambda\}$ and $0\leq \lambda \leq 1$ . We also give a reverse for the classical numerical radius power inequality $w(A^{n})\leq w^{n}(A)$ for any operator $A \in B(H)$ in the case when $n=2.$

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