Notes on the numerical radius for adjointable operators on Hilbert $C^*$-modules
J. Li, K. Wu, Q. Xu
TL;DR
This work extends numerical-radius theory to adjointable operators on Hilbert $C^*$-modules by introducing the spatial numerical radius $\tilde w(T)$ and proving the fundamental identity $\tilde w(T)=\sup_{\theta\in[0,2\pi]} \|\mathrm{Re}(e^{i\theta}T)\|$. It then shows $\tilde w(T)\le w(T)$ and establishes a faithful-representation correspondence $\tilde w(T)=w(\pi(T))\le w(T)$, enabling a normal-operator result: $w(T)=\|T\|$ for normal $T$. The paper also provides counterexamples showing that two standard equalities from the Hilbert-space setting, namely $w(T)=\sup_{\theta} \|\mathrm{Re}(e^{i\theta}T)\|$ and $w(T)=\tfrac{1}{2}\|T\|$ when $T^2=0$, can fail in the module context. It further derives new inequalities involving $\tilde w(T)$ and a spectral-radius bound for sums of products, illustrating the utility of the approach for establishing operator-inequality results in this generalized setting.
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 $\sup\limits_{θ\in [0,2π]}\|\mbox{Re}(e^{iθ}T_2)\|<w(T_2)$. In addition, a new characterization of the spatial numerical radius is given, and it is proved that $w\big(π(T)\big)\le w(T)$ for every faithful representation $(π, X)$ of $\mathcal{L}(H)$ and every $T\in\mathcal{L}(H)$. Some inequalities are derived based on the newly obtained results.