Some refinements of numerical radius inequalities

TL;DR

This paper refines existing inequalities related to the numerical radius of operators, especially for hyponormal operators, and provides a reverse inequality for the power inequality when n=2.

Contribution

It introduces new bounds for the numerical radius using the Kantorovich constant and refines classical inequalities for specific operator classes.

Findings

Refined inequality for the numerical radius of hyponormal operators.

Established a reverse inequality for the case n=2 in the power inequality.

Provided bounds involving the Kantorovich constant and operator norms.

Abstract

In this paper, we give some refinements for the second inequality in $\frac{1}{2}\|A\| \leq w(A) \leq \|A\|$, 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 $w(A)\leq \dfrac{1}{\displaystyle {2\inf_{\| x \|=1}}\zeta(x)}\| |A|+|A^{*}|\|\leq \dfrac{1}{2}\| |A|+|A^*|\|$, where $\zeta(x)=K(\frac{\langle |A|x,x \rangle}{\langle |A^{*}|x,x \rangle},2)^{r},~~~r=\min\{\lambda,1-\lambda\}$ 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 $n=2$.