Measures of noncompactness in Hilbert $C^{\ast }$-modules

Abstract

Consider a countably generated Hilbert $C^{\ast }$-module $\mathcal{M}$ over a $C^{\ast }$-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 ${\chi }^{\ast }$. We obtain $\lambda \equiv {\chi }^{\ast }$. Related inequalities involving other known measures of noncompactness, e.g. Kuratowski and Istrăţescu are laso obtained as well as some related results on adjontable operators.

14 pages