The Kaczmarz Algorithm in Hilbert $C^{*}$-modules

TL;DR

This paper extends the classical Kaczmarz algorithm to Hilbert $C^*$-modules, enabling stable recovery of continuous families of vectors and establishing conditions for generating frames via the algorithm.

Contribution

It introduces a novel extension of the Kaczmarz algorithm to Hilbert $C^*$-modules and develops tools for frame generation and vector recovery in this setting.

Findings

The algorithm effectively recovers vectors in Hilbert $C^*$-modules.

Continuous families of vectors can be uniformly recovered.

Conditions are provided for frames generated by the Kaczmarz algorithm.

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.