Extensions of the Hilbert-multi-norm in Hilbert $C^*$-modules

Abstract: Dales and Polyakov introduced a multi-norm $\left( \left|\cdot\right|_n^{(2,2)}:n\in\mathbb{N}\right)$ based on a Banach space $\mathscr{X}$ and showed that it is equal with the Hilbert-multi-norm $\left( \left|\cdot\right|_n^{\mathscr{H}}:n\in\mathbb{N}\right)$ based on an infinite-dimensional Hilbert space $\mathscr{H}$. We enrich the theory and present three extensions of the Hilbert-multi-norm for a Hilbert $C^*$-module $\mathscr{X}$. We denote these multi-norms by $\left( \left|\cdot\right|_n^{\mathscr{X}}:n\in\mathbb{N}\right)$, $\left( \left|\cdot\right|_n^{*}:n\in\mathbb{N}\right)$, and $\left( \left|\cdot\right|_n^{\mathcal{P}\left(\mathfrak{A} \right) }:n\in\mathbb{N}\right)$. We show that $\left|x\right|_n^{\mathcal{P}\left(\mathfrak{A} \right) }\geq\left|x\right|_n^{\mathscr{X}}\leq \left|x\right|_n^{*}$ for each $x\in\mathscr{X}^n$. In the case when $\mathscr{X}$ is a Hilbert $\mathbb{K}\left(\mathscr{H}\right)$-module, for each $x\in\mathscr{X}^n$, we observe that $\left|\cdot\right|_n^{\mathcal{P}\left(\mathfrak{A} \right)}=\left|\cdot\right|_n^{\mathscr{X}}$. Furthermore, if $\mathscr{H}$ is separable and $\mathscr{X}$ is infinite-dimensional, we prove that $\left|x\right|_n^{\mathscr{X}}=\left|x\right|_n^{*}$. Among other things, we show that small and orthogonal decompositions with respect to these multi-norms are equivalent. Several examples are given to support the new findings.