Power-norms based on Hilbert $C^*$-modules

TL;DR

This paper develops a new power-norm for Hilbert $C^*$-modules, introduces a novel class of absolutely $(2,2)$-summing operators, and explores their properties, linking them to Hilbert--Schmidt operators in specific module contexts.

Contribution

It introduces a new power-norm on Hilbert $C^*$-modules and defines a new class of absolutely $(2,2)$-summing operators, connecting them to existing Hilbert--Schmidt operator classes.

Findings

The power-norm $ orm{ullet}_n^{ ext{E}}$ has fundamental properties.

The class $ ilde{ extPi}_2( ext{E}, ext{F})$ is introduced and studied.

For countably generated modules over unital commutative $C^*$-algebras, $ ilde{ extPi}_2( ext{E})$ coincides with the Hilbert--Schmidt operators.

Abstract

Suppose that $\mathscr{E}$ and $\mathscr{F}$ are Hilbert $C^*$-modules. We present a power-norm $\left(\left\|\cdot\right\|^{\mathscr{E}}_n:n\in\mathbb{N}\right)$ based on $\mathscr{E}$ and obtain some of its fundamental properties. We introduce a new definition of the absolutely $(2,2)$-summing operators from $\mathscr{E}$ to $\mathscr{F}$, and denote the set of such operators by $\tilde{\Pi}_2(\mathscr{E},\mathscr{F})$ with the convention $\tilde{\Pi}_2(\mathscr{E})=\tilde{\Pi}_2(\mathscr{E},\mathscr{E})$. It is known that the class of all Hilbert--Schmidt operators on a Hilbert space $\mathscr{H}$ is the same as the space $\tilde{\Pi}_2(\mathscr{H})$. We show that the class of Hilbert--Schmidt operators introduced by Frank and Larson coincides with the space $\tilde{\Pi}_2(\mathscr{E})$ for a countably generated Hilbert $C^*$-module $\mathscr{E}$ over a unital commutative $C^*$-algebra. These results motivate us to investigate the properties of the space $\tilde{\Pi}_2(\mathscr{E},\mathscr{F})$.