Numerical radius inequalities for certain operator matrices

In this paper, we establish new upper and lower bounds for the numerical radii of certain operator matrices, which generalize and improve on existing ones. Also, we prove that for an arbitrary operator A is an element of B(H), alpha, beta is an element of (0, 1) and r >= 2 max{alpha, beta, 1 - alpha, 1 - beta}, w(2r)(A) <= min{a, b}, where a = parallel to beta/2 vertical bar A vertical bar(r/beta) + 1 - beta/2 vertical bar A*vertical bar(r/1-beta)parallel to + 1/2 w(r)(A(2)) and b = parallel to alpha(2)vertical bar A vertical bar(r/alpha) + (1 - alpha)(2)vertical bar A*vertical bar(r/1-alpha)parallel to + 2 alpha(1 - alpha)parallel to R(vertical bar A vertical bar(r/2 alpha)vertical bar A*vertical bar(r/2(1-alpha))parallel to. Here w(.) and parallel to.parallel to are the numerical radius and the usual operator norm, respectively. Also, A*, vertical bar A vertical bar and R(A) denote the adjoint, the absolute value and the real part of A, respectively. It is worth to point out here that giving specific values for alpha, beta is an element of (0, 1) gives accurate estimates for the numerical radius.