Tightening Bounds on the Numerical Radius for Hilbert Space Operators

TL;DR

This paper establishes a new lower bound for the numerical radius of accretive and dissipative operators on Hilbert spaces, improving understanding of their spectral properties.

Contribution

It introduces a tighter lower bound for the numerical radius of accretive and dissipative operators, linking it to the operator norm with a specific constant.

Findings

The numerical radius is at least rac{ oot 3 3}{3} times the operator norm for accretive operators.

The same bound applies to dissipative operators in the Hilbert space setting.

This result refines previous bounds and enhances spectral analysis tools.

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 \[\frac{\sqrt{3}}{3}\left\| S \right\|\le \omega \left( S \right),\] where $\left\| \cdot \right\| $ and $\omega \left( \cdot \right)$ denote the operator norm and the numerical radius, respectively.