On summability, multipliability, product integrability and parallel translation

TL;DR

This paper establishes necessary and sufficient conditions for various product integrals of step and right regulated mappings in Banach algebras, generalizing classical results and connecting to differential equations and parallel translation.

Contribution

It introduces new conditions for product integrals in Banach algebras and generalizes summability and multipliability concepts to well-ordered index sets.

Findings

Characterization of product integrals for step mappings

Generalization of infinite series and product results

Applications to differential equations and parallel translation

Abstract

In this paper we provide necessary and sufficient conditions for the existence of the Kurzweil, McShane and Riemann product integrals of step mappings with well-ordered steps, and for right regulated mappings with values in Banach algebras. Our basic tools are the concepts of summability and multipliability of families in normed algebras indexed by well-ordered subsets of the real line. These concepts also lead to the generalization of some results from the usual theory of infinite series and products. Finally, we present two applications of product integrals: First, we describe the relation between Stieltjes-type product integrals, Haahti products, and parallel translation operators. Second, we provide a link between the theory of strong Kurzweil product integrals and strong solutions of linear generalized differential equations.