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

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 $w(T)=\sup \{\mathrm{\parallel }⟨Tx,x⟩\mathrm{\parallel }:x\in H,\mathrm{\parallel }x\mathrm{\parallel }=1\}$. It is proved that $w(T)=\mathrm{\parallel }T\mathrm{\parallel }$ 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 $w(T_1)\mathrm{\ne }\frac{1}{2}\mathrm{\parallel }{T}_1\mathrm{\parallel }$ and $\underset{\theta \in [0,2\pi ]}{\sup }\mathrm{\parallel }Re(e^{i\theta }{T}_2)\mathrm{\parallel }<w(T_2)$. In addition, a new characterization of the spatial numerical radius is given, and it is proved that $w(\pi (T))\le w(T)$ 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.