Numerical radius in Hilbert $C^*$-modules

TL;DR

This paper introduces a new numerical radius norm for elements in Hilbert $C^*$-modules using linking algebra, establishing its properties and relations to the standard norm.

Contribution

It defines a novel numerical radius norm on Hilbert $C^*$-modules and explores its properties, including bounds, equivalence conditions, and a refined triangle inequality.

Findings

$ ext{Omega}(ullet)$ is a norm on the module.

Bounds: $rac{1}{2} orm{x} \,\leq\, \Omega(x) \leq \norm{x}$.

Refined triangle inequality for $ ext{Omega}(ullet)$.

Abstract

Utilizing the linking algebra of a Hilbert $C^*$-module $\big(\mathscr{V}, {\|\!\cdot\!\|}\big)$, we introduce $\Omega(x)$ as a definition of numerical radius for an element $x\in\mathscr{V}$ and then show that $\Omega(\cdot)$ is a norm on $\mathscr{V}$ such that $\frac{1}{2}{\|x\|} \leq \Omega(x) \leq {\|x\|}$. In addition, we obtain an equivalent condition for $\Omega(x) = \frac{1}{2}{\|x\|}$. Moreover, we present a refinement of the triangle inequality for the norm $\Omega(\cdot)$. Some other related results are also discussed.