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

Abstract

Several upper and lower bounds for the numerical radius of 2 × 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, thenwhere 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 0 B C 0 . As application of results obtained, we show that if B, C are self-adjoint operators then, max B + C 2 , B − C 2 ≤ B 2 + C 2 + 2w(|B||C|).