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

TL;DR

This paper extends the concept of Hilbert-multi-norms to Hilbert $C^*$-modules, establishing inequalities, equalities in special cases, and demonstrating the equivalence of certain decompositions, supported by multiple examples.

Contribution

The paper introduces three new multi-norms for Hilbert $C^*$-modules and explores their relationships, including conditions for equality and equivalence of decompositions, enriching the existing theory.

Findings

$ orm{x}_n^{rak{A}} ext{ is greater or equal to } orm{x}_n^{ ext{X}}$ and less or equal to $ orm{x}_n^{*}$.

In Hilbert $rak{K}(rak{H})$-modules, the multi-norms $ orm{ullet}_n^{rak{A}}$ and $ orm{ullet}_n^{ ext{X}}$ coincide.

For separable $rak{H}$ and infinite-dimensional $ ext{X}$, $ orm{ullet}_n^{ ext{X}}$ equals $ orm{ullet}_n^{*}$.

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.