The weighted numerical radius in Hilbert C*-modules

Abstract

In this paper, we introduce the definition of the weighted numerical radius $\Omega(x)$ for $x\in \mathcal{E}$ by using the linking algebra of a Hilbert C$^{*}$-modules $\mathcal{E}$, which extends the definition of numerical radius $\Omega(x)$ given by Zamani [ Math. Inequal. Appl. 24 (2021), 1017-1030 ]. Among other results, we show that $\Omega_{\nu}( x)$ is a norm on $\mathcal{E}$ such that
$$\frac{1}{2}\|x\|_{\mathcal{E}}\leq \max \{ \nu, 1-\nu\}\|x\|_{\mathcal{E}}\leq \Omega_{\nu}( x) \leq \|x\|_{\mathcal{E}},$$ where $0\leq \nu\leq1$. In addition, some relevant results are discussed.