Year: 2025
Type: preprint
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)\|
Source: arXiv (Cornell University)
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4416150642
Cites:
Cited by:
Related to: