Measures of noncompactness on the standard Hilbert $C^*$-module

TL;DR

This paper introduces a new measure of noncompactness for Hilbert $C^*$-modules and compares it with existing measures, enhancing understanding of compactness in this mathematical setting.

Contribution

It defines a novel measure of noncompactness on Hilbert $C^*$-modules and explores its properties and relationships with existing measures.

Findings

The new measure $mbda$ characterizes $$-precompact sets.

Relationships between $mbda$ and classical measures are established.

Properties of all measures are analyzed in the context of Hilbert $C^*$-modules.

Abstract

We define a measure of noncompactness $\lambda$ on the standard Hilbert $C^*$-module $l^2(\mathcal A)$ over a unital $C^*$-algebra, such that $\lambda(E)=0$ if and only if $E$ is $\mathcal A$-precompact (i.e.\ it is $\varepsilon$-close to a finitely generated projective submodule for any $\varepsilon>0$) and derive its properties. Further, we consider the known, Kuratowski, Hausdorff and Istr\u{a}\c{t}escu measure of noncomapctnes on $l^2(\mathcal A)$ regarded as a locally convex space with respect to a suitable topology, and obtain their properties as well as some relationship between them and introduced measure of noncompactness $\lambda$.