The Kaczmarz Algorithm in Hilbert C*-modules

Authors

Daniel Alpay, Chapman UniversityFollow
Chad Berner, Missouri University of Science and Technology
Eric S. Weber, Iowa State University

Document Type

Article

Publication Date

11-25-2025

Abstract

The Kaczmarz algorithm in Hilbert spaces is a classical iterative method for stably recovering vectors from inner product data. In this paper, we extend the algorithm to the setting of Hilbert C*-modules and establish analogues of its effectiveness in both finite-dimensional and stationary cases. Consequently, we demonstrate that continuous families of elements in a Hilbert space can be uniformly recovered using the Kaczmarz algorithm. Additionally, we develop a normalized Cauchy transform for continuous families of measures and use it to provide sufficient conditions under which standard frames in Hilbert C(X)-modules can be generated by the Kaczmarz algorithm and realized as orbits of bounded operators.

Comments

This article was originally published in Expositiones Mathematicae, volume 44, issue 2, in 2026. https://doi.org/10.1016/j.exmath.2025.125738

Recommended Citation

Daniel Alpay, Chad Berner, Eric S. Weber, The Kaczmarz Algorithm in Hilbert C*-modules. Expositiones Mathematicae 44 (2) (2026), https://doi.org/10.1016/j.exmath.2025.125738, 125738.

Peer Reviewed

1

Copyright

The authors

Creative Commons License

This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.