Measures of noncompactness in Hilbert $C^\ast$-modules
Abstract
Consider a countably generated Hilbert $C^\ast$-module $\mathcal{M}$ over a unital $C^\ast$-algebra $\mathcal{A}$. There is a measure of noncompactness $\lambda$ defined, roughly as the distance from finitely generated sets, 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 introduced 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 also obtained, as well as, some related results on adjointable operators.
Fields of Study & Subjects
Advanced Operator Algebra Research
Nonlinear Differential Equations Analysis
Advanced Banach Space Theory
Publication details
- Journal / source
- Publicationes Mathematicae Debrecen
- Publisher
- University of Debrecen/ Debreceni Egyetem
- Volume
- 108
- Issue
- 3-4
- Pages
- 327–354
- Publication date
- 1 April 2026