General Upper Bounds for the Numerical Radii of Hilbert Space Operators

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 𝐴 is a bounded linear operator on a complex Hilbert space, then 𝑤 2𝑟 (𝐴) ≤ 1+𝛼 8 ‖|𝐴| 2𝑟 +|𝐴 ∗ | 2𝑟‖+ 1+𝛼 4 𝑤(|𝐴| 𝑟 |𝐴 ∗ | 𝑟 )+ 1−𝛼 2 𝑤 𝑟 (𝐴 2 ) for every r ≥ 1 and α ∈ 0,1. This substantially improves on the existing inequality 𝑤 2𝑟 (𝐴) ≤ 1 2 ‖|𝐴| 2𝑟 + |𝐴 ∗ | 2𝑟‖. Here 𝑤(. ) and ||. || denote the numerical radius and the usual operator norm, respectively.