SOME REFINEMENTS OF NUMERICAL RADIUS INEQUALITIES

We propose some refinements for the second inequality in 1/2 parallel to A parallel to <= w(A) parallel to A parallel to, where A epsilon B(H). In particular, if A is hyponormal, then, by refining the Young inequality with the Kantorovich constant K(center dot, center dot), we show that w(A) <= 1/2 inf parallel to x parallel to=1 zeta(x) k vertical bar A vertical bar + vertical bar A vertical bar k. 1 2 k vertical bar A vertical bar + vertical bar A vertical bar k, where zeta(x) = K. h vertical bar A vertical bar x, xi h vertical bar A vertical bar x, xi, 2. r, r = min{lambda, 1 -lambda}, and 0 <= lambda <= 1. We also give a reverse for the classical numerical radius power inequality w(A(n)). wn(A) for any operator A epsilon B(H) in the case where n = 2.