Numerical radius in Hilbert $C^*$-modules

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.