New Estimates for the Numerical Radius

TL;DR

This paper introduces new inequalities for the numerical radius of operators in Hilbert spaces, refining existing bounds and enabling broader generalizations with applications to accretive-dissipative operators.

Contribution

It presents novel inequalities for the numerical radius, including refinements for accretive-dissipative operators, expanding the theoretical understanding of operator bounds.

Findings

New inequalities for the numerical radius of operator sums

Refinement of the inequality for accretive-dissipative operators

Enhanced bounds relating numerical radius and operator norm

Abstract

In this article, we present new inequalities for the numerical radius of the sum of two Hilbert space operators. These new inequalities will enable us to obtain many generalizations and refinements of some well known inequalities, including multiplicative behavior of the numerical radius and norm bounds. Among many other applications, it is shown that if $T$ is accretive-dissipative, then \[\frac{1}{\sqrt{2}}\left\| T \right\|\le \omega \left( T \right),\] where $\omega \left( \cdot \right)$ and $\left\| \cdot \right\|$ denote the numerical radius and the usual operator norm, respectively. This inequality provides a considerable refinement of the well known inequality $\frac{1}{2}\|T\|\leq \omega(T).$