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

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.