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 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.

Keywords

• Numerical radius
• Usual operator norm
• Operator matrix
• Buzano
• s inequality.

References

1. Al-Dolat, M. (2024). General upper bounds for the numerical radii of Hilbert space operators The Eurasia Proceedings of Science, Technology, Engineering & Mathematics (EPSTEM), 28, 375-381.