Numerical Radius Bounds via the Euclidean Operator Radius and Norm

TL;DR

This paper introduces new bounds for the numerical radius of operators using a generalized Cauchy-Schwarz inequality, supported by numerical examples demonstrating their effectiveness and extending existing results in operator theory.

Contribution

It presents novel bounds for the numerical radius based on the Euclidean operator radius and norm, generalizing classical inequalities and providing extensive numerical validation.

Findings

New bounds for the numerical radius are established.

Over 15 numerical examples demonstrate the bounds' advantages.

A characterization of accretive-dissipative operators relating Euclidean norm and numerical radius.

Abstract

In this paper, we begin by showing a new generalization of the celebrated Cauchy-Schwarz inequality for the inner product. Then, this generalization is used to present some bounds for the Euclidean operator radius and the Euclidean operator norm. These bounds will be used then to obtain some bounds for the numerical radius in a way that extends many well-known results in many cases. The obtained results will be compared with the existing literature through numerical examples and rigorous approaches, whoever is applicable. In this context, more than 15 numerical examples will be given to support the advantage of our findings. Among many consequences, will show that if $T$ is an accretive-dissipative bounded linear operator on a Hilbert space, then ${{\left\| \left( \Re T,\Im T \right) \right\|}_{e}}=\omega \left( T \right)$, where $\omega(\cdot), \|(\cdot,\cdot)\|_e, \Re T$ and $\Im T$ denote, respectively, the numerical radius, the Euclidean norm, the real part and the imaginary part.