Furtherance of numerical radius inequalities of Hilbert space operators

Abstract

If A, B are bounded linear operators on a complex Hilbert space, then we prove that

$$\begin{aligned} w(A)\le & {} \frac{1}{2}\left( \Vert A\Vert +\sqrt{r\left( |A||A^*|\right) }\right) ,\\ w(AB \pm BA)\le & {} 2\sqrt{2}\Vert B\Vert \sqrt{ w^2(A)-\frac{c^2(\mathfrak {R}(A))+c^2(\mathfrak {I}(A))}{2} }, \end{aligned}$$

where $w(\cdot ),\left\| \cdot \right\|$, and $r(\cdot )$ are the numerical radius, the operator norm, the Crawford number, and the spectral radius respectively, and $\mathfrak {R}(A)$, $\mathfrak {I}(A)$ are the real part, the imaginary part of A respectively. The inequalities obtained here generalize and improve on the existing well known inequalities.