Refinements of norm and numerical radius inequalities

Abstract

Several refinements of norm and numerical radius inequalities of bounded linear operators on a complex Hilbert space are given. In particular, we show that if $A$ is a bounded linear operator on a complex Hilbert space, then

$$\frac{1}{4}‖A^{*}A+A{A}^{*}‖\le \frac{1}{8}\left({‖A+A^{*}‖}^2+{‖A-A^{*}‖}^2+c^2(A+A^{*})+c^2(A-A^{*})\right)\le w^2(A)$$

and

$$\frac{1}{2}‖A^{\ast }A+A{A}^{\ast }‖-\frac{1}{4}{‖{(A+A^{\ast })}^2{(A-A^{\ast })}^2‖}^{\frac{1}{2}}\le w^2(A)\le \frac{1}{2}‖A^{\ast }A+A{A}^{\ast }‖,$$

where $\parallel \cdot \parallel$, $w(\cdot )$ and $c(\cdot )$ are the operator norm, the numerical radius and the Crawford number, respectively. Further, we prove that if $A,D$ are bounded linear operators on a complex Hilbert space, then

$$‖A{D}^{\ast }‖\le {‖{\displaystyle {\int }_0^1{\left((1-t)({\left|A\right|}^2+{\left|D\right|}^2)/2+t‖A{D}^{\ast }‖I\right)}^{2}dt}‖}^{\frac{1}{2}}\le \frac{1}{2}{‖{\left|A\right|}^2+D‖}^2,$$

where $|A{|}^2=A^{\ast }A$ and $|D{|}^2=D^{\ast }D$. This is a refinement of a well-known inequality obtained by Bhatia and Kittaneh.