Notes on the numerical radius for adjointable operators on Hilbert $C^*$-modules

TL;DR

This paper investigates the properties of the numerical radius for adjointable operators on Hilbert $C^*$-modules, establishing when it equals the operator norm and providing new characterizations and inequalities.

Contribution

It introduces new characterizations of the numerical radius, explores its relation to the operator norm, and presents examples showing differences in specific cases.

Findings

w(T)=‖T‖ for normal operators

Counterexamples where w(T)≠‖T‖/2

A new characterization of the spatial numerical radius

Abstract

Given a Hilbert module $H$ over a $C^*$-algebra, let $\mathcal{L}(H)$ be the set of all adjointable operators on $H$. For each $T\in\mathcal{L}(H)$, its numerical radius is defined by $w(T)=\sup\big\{\|\langle Tx, x \rangle\|: x\in H, \|x\|=1\big\}$. It is proved that $w(T)=\|T\|$ whenever $T$ is normal. Examples are constructed to show that there exist Hilbert module $H$ over certain $C^*$-algebra and $T_1,T_2\in \mathcal{L}(H)$ with $T_1^2=0$ such that $w(T_1)\ne \frac12 \|T_1\|$ and $\sup\limits_{\theta\in [0,2\pi]}\|\mbox{Re}(e^{i\theta}T_2)\|<w(T_2)$. In addition, a new characterization of the spatial numerical radius is given, and it is proved that $w\big(\pi(T)\big)\le w(T)$ for every faithful representation $(\pi, X)$ of $\mathcal{L}(H)$ and every $T\in\mathcal{L}(H)$. Some inequalities are derived based on the newly obtained results.