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

Abstract

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 + ∝ / 4 |||T|^2r  + |T*|^2r|| + 1 + ∝ / 2 w^r (T²) for every r ≥ 1 and ∝ ∈ [0.1] .

This substantially improves on the existing inequality w^2r (T) 1/2 |||T|^2r + |T*|^2r||. Here w(.) and ||.|| denote the numerical radius and the usual operator norm, respectively.