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 $|.|$ 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 $\left[\begin{array}{cc} 0 & B C& 0 \end{array}\right]$. As application of results obtained, we show that if $B,C$ are self-adjoint operators then, $\max \Big {|B+C|^2 , |B-C|^2 \Big}\leq \left |B^2+C^2 \right |+2w(|B||C|). $