Upper bounds for the numerical radii of powers of Hilbert space operators

We present several upper bounds for the numerical radii of 2x2 operator matrices. We employ these bounds to improve on some known numerical radius inequalities for powers of Hilbert space operators. In particular, we show that if T is a bounded linear operator on a complex Hilbert space, then w(2r)(T) <= 1+alpha/4 parallel to vertical bar T vertical bar(2r) + vertical bar T*vertical bar(2r)parallel to +1-alpha/2 w(r)(T-2) for every r >= 1and alpha is an element of [0,1]. This substantially improves on the existing inequality w(2r) (T) <= 1/2 parallel to vertical bar T vertical bar(2r) + vertical bar T*vertical bar(2r)parallel to. Here w((center dot)) and parallel to(center dot)parallel to denote the numerical radius and the usual operator norm, respectively.