Numerical radius inequalities of $2\times 2$ operator matrices

Abstract

Several upper and lower bounds for the numerical radius of $2\times 2$ operator matrices are developed which refine and generalize the earlier related bounds. In particular, we show that if $B,C$ are bounded linear operators on a complex Hilbert space, then \begin{eqnarray*} && \frac{1}{2}\max \left \{ \|B\|, \|C\| \right \}+\frac{1}{4} \left | \|B+C^*\|-\|B-C^*\| \right | &&\leq w \left(\left[\begin{array}{cc} 0 & B C& 0 \end{array}\right]\right)\\ &&\leq \frac{1}{2} \max \left\{\|B\|,\|C\|\right \}+\frac{1}{2}\max \left \{r^{\frac{1}{2}}(|B||C^*|),r^{\frac{1}{2}}(|B^*||C|)\right\}, \end{eqnarray*} where $w(.)$, $r(.)$ and $\Vert .\Vert$ are the numerical radius, spectral radius and operator norm of a bounded linear operator, respectively. We also obtain equality conditions for the numerical radius of the operator matrix $\left[\begin{array}{ccc}0& BC& 0\end{array}\right]$. As application of results obtained, we show that if $B,C$ are self-adjoint operators then, $\max \{\Vert B+C{\Vert }^2,\Vert B-C{\Vert }^2\}\le \Vert B^2+C^2\Vert +2w(|B||C|).$

16 pages

References in corpus (1)

• Furtherance of Numerical radius inequalities of Hilbert space operators