Measures of noncompactness in Hilbert $C^*$-modules

TL;DR

This paper introduces and compares measures of noncompactness in Hilbert $C^*$-modules, establishing their equivalence and exploring related inequalities and properties of adjointable operators.

Contribution

It defines a measure of noncompactness in Hilbert $C^*$-modules and proves its equivalence to the Hausdorff measure, extending the understanding of noncompactness in this setting.

Findings

The measure $ ext{lambda}$ is equivalent to the Hausdorff measure $ ext{chi}^*$.

Derived inequalities involving Kuratowski and Istrțescu measures.

Results on properties of adjointable operators in this context.

Abstract

Consider a countably generated Hilbert $C^*$-module $\mathcal M$ over a $C^*$-algebra $\mathcal A$. There is a measure of noncompactness $\lambda$ defined, roughly as the distance from finitely generated projective submodules, which is independent of any topology. We compare $\lambda$ to the Hausdorff measure of noncompactness with respect to the family of seminorms that induce a topology recently iontroduced by Troitsky, denoted by $\chi^*$. We obtain $\lambda\equiv\chi^*$. Related inequalities involving other known measures of noncompactness, e.g. Kuratowski and Istr\u{a}\c{t}escu are laso obtained as well as some related results on adjontable operators.