Vol. 51, No. 1, 1974

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Product integrals and inverses in normed rings

Jon Craig Helton

Vol. 51 (1974), No. 1, 155–166

This paper concerns product integrals of functions with values in a normed complete ring. The inverses of elements obtained as such integrals are investigated. In particular, the conditions under which [x y(1 + G)]1 exists are shown to be related to the requirement that xy|G2| = 0. Since the existence of [x y(1 + G)]1 is connected with the existence of the product integrals y x(1 + G) and x y(1 G), the study of the inverse leads to a study of the conditions under which these integrals exist when x y(1 + G) is known to exist. Commutative and noncommutative rings are considered.

Mathematical Subject Classification 2000
Primary: 46H99
Secondary: 46G99
Received: 31 October 1972
Published: 1 March 1974
Jon Craig Helton