Some refinements of numerical radius inequalities

• Date: 2020
• Main Authors: Heydarbeygi, Z., Amyari, M., Khanehgir, M., A.
• Format: Article
• Language: English
• Published: Institute of Mathematics, NAS of Ukraine 2020
• Online Access: https://umj.imath.kiev.ua/index.php/umj/article/view/6027
• Journal Title: Ukrains’kyi Matematychnyi Zhurnal

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 $n=2.$