General Upper Bounds for the Numerical Radii of Hilbert Space Operators

Abstract

We present a collection upper bounds for the numerical radii of a certain 2 × 2 operator matrices. We use these bounds to improve on some known numerical radius inequalities for powers of Hilbert space operators. In particular, we show that if ???? is a bounded linear operator on a complex Hilbert space, then ???? 2???? (????) ≤ 1+???? 8 ‖|????| 2???? +|???? ∗ | 2????‖+ 1+???? 4 ????(|????| ???? |???? ∗ | ???? )+ 1−???? 2 ???? ???? (???? 2 ) for every r ≥ 1 and α ∈ [0,1]. This substantially improves on the existing inequality ???? 2???? (????) ≤ 1 2 ‖|????| 2???? + |???? ∗ | 2????‖. Here ????(. ) and ||. || denote the numerical radius and the usual operator norm, respectively.