Notes on the numerical radius for adjointable operators on Hilbert $C^{\ast }$-modules

Abstract

Given a Hilbert module $H$ over a $C^{\ast }$-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 $w(T)=\sup \{\Vert ⟨Tx,x⟩\Vert :x\in H,\Vert x\Vert =1\}$. It is proved that $w(T)=\Vert T\Vert$ whenever $T$ is normal. Examples are constructed to show that there exist Hilbert module $H$ over certain $C^{\ast }$-algebra and $T_1,T_2\in \mathcal{L}(H)$ with $T_1^2=0$ such that $w(T_1)\ne \frac{1}{2}\Vert T_1\Vert$ and $\sup\limits_{θ\in [0,2π]}\|\mbox{Re}(e^{iθ}T_2)\|<w(T_2)$. In addition, a new characterization of the spatial numerical radius is given, and it is proved that $w(\ddot{I}€(T))\le w(T)$ for every faithful representation $(\ddot{I}€,X)$ of $\mathcal{L}(H)$ and every $T\in \mathcal{L}(H)$. Some inequalities are derived based on the newly obtained results.