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Multiplicative Integration of Infinite Products

Published online by Cambridge University Press:  20 November 2018

David Lowell Lovelady
David Lowell Lovelady*
Affiliation:
Georgia Institute of Technology, Atlanta, Georgia University of South Carolina, Columbia, South Carolina
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Let G be a complete normed abelian group with norm N1. Let S be an interval (bounded or otherwise) of real numbers. We propose to study the Stieltjes integral equation

where p is in G, a is in S, and each Fk is a function on S each value of which is a function from G to G. Our primary tools of investigation will be the works of J. S. MacNerney [6; 7] and their extensions by the author [4; 5]. Our main result, Theorem 4, will show that the equation above can be solved by a product integral of infinite products of solutions for the summands of the integrator. After obtaining our results, we shall specialize them to a linear situation and then show how this specialization allows us to obtain representations for analytic functions having only invertible values in a complex Banach algebra with identity.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 4 , August 1971 , pp. 692 - 698
Copyright
Copyright © Canadian Mathematical Society 1971

References

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