Some refinements of numerical radius inequalities

Abstract

In this paper, we give some refinements for the second inequality in $\frac{1}{2}\Vert A\Vert \le w(A)\le \Vert A\Vert$, 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 $w(A)\le \frac{1}{2\underset{\Vert x\Vert =1}{\inf }\zeta (x)}\Vert |A|+|A^{\ast }|\Vert \le \frac{1}{2}\Vert |A|+|A^{\ast }|\Vert$, where $\zeta (x)=K(\frac{⟨|A|x,x⟩}{⟨|A^{\ast }|x,x⟩},2{)}^r,r=\min \{\lambda ,1-\lambda \}$ and $0\le \lambda \le 1$ . We also give a reverse for the classical numerical radius power inequality $w(A^n)\le w^n(A)$ for any operator $A\in B(H)$ in the case when $n=2$.