INEQUALITIES FOR THE NORM AND NUMERICAL RADIUS FOR HILBERT 𝐶 * -MODULE OPERATORS

Abstract

In this paper, we introduce some inequalities between the operator norm and the numerical radius of adjointable operators on Hilbert 𝐶*-module spaces. Moreover, we establish some new refinements of numerical radius inequalities for Hilbert space operators. More precisely, we prove that if 𝑇 ∈ 𝐵(𝐻) and min (︁‖𝑇 + 𝑇*‖^2/2, ‖𝑇 − 𝑇*‖^2/2)︁ ≤ max (︁inf_‖𝑥‖=1 ‖𝑇𝑥‖^2, inf_‖𝑥‖=1 ‖𝑇*𝑥‖^2)︁, then ‖𝑇‖≤ 2^(1/2)𝜔(𝑇); this is a considerable improvement of the classical inequality ‖𝑇‖≤ 2𝜔(𝑇).

Keywords

• bounded linear operator
• hilbert space
• norm inequality
• numerical radius