Numerical Radius Inequalities for Certain 2 x 2 Operator Matrices

We prove several numerical radius inequalities for certain 2x2 operator matrices. Among other inequalities, it is shown that if X, Y, Z, and W are bounded linear operators on a Hilbert space, then w([X Y Z W]) >= max (w(X), w(W), w(Y + Z)/2, w(Y - Z/2)) and w([X Y Z W]) <= max (w(X), w(W)) + w(Y + Z) + w(Y - Z/2. As an application of a special case of the second inequality, it is shown that parallel to X parallel to/2 + vertical bar parallel to Re X parallel to - parallel to x parallel to/2 vertical bar/4 + vertical bar parallel to Im X parallel to - parallel to x parallel to/2 vertical bar/4 <= w(X), which is a considerable improvement of the classical inequality parallel to x parallel to/2 <= w(X). Here w(.) and parallel to.parallel to are the numerical radius and the usual operator norm, respectively.