Tightening Bounds on the Numerical Radius for Hilbert Space Operators

TL;DR

This paper addresses tightening the relationship between the numerical radius $\omega(S)$ and the operator norm $\|S\|$ for bounded Hilbert-space operators. It introduces a functional-calculus bound using nonnegative functions $f,g$ with $f(t)g(t)=t$ to bound $\omega^2(S)$, and extends these ideas to $2\times 2$ block operator matrices to derive sharper upper bounds for $\omega(S)$. A key structural result shows that $\|S\|$ can be controlled by $\omega(S)$ and the spectral radius $r(|\Re S|\,|\Im S|)$, and this leads to a new lower bound $\frac{\sqrt{3}}{3}\|S\| \le \omega(S)$ when $S$ is accretive or dissipative. Collectively, these results improve classical Kittaneh-type inequalities and provide sharper, scenario-specific estimates that enhance our understanding of the numerical radius in operator theory.

Abstract

Let $S$ be a bounded linear operator on a Hilbert space. We show that if $S$ is accretive (resp. dissipative the sense that $\frac{S-{{S}^{*}}}{2i}$ is positive) in the sense that $\frac{S+{{S}^{*}}}{2}$ is positive, then \[\frac{\sqrt{3}}{3}\left\| S \right\|\le ω\left( S \right),\] where $\left\| \cdot \right\| $ and $ω\left( \cdot \right)$ denote the operator norm and the numerical radius, respectively.

Theorems & Definitions (42)

• Lemma 1.1
• Lemma 1.2
• Lemma 1.3
• Lemma 1.4
• Lemma 1.5
• Lemma 1.6
• Lemma 1.7
• Lemma 1.8
• Theorem 2.1
• proof