Product integrals and exponentials in commutative Banach algebras

Author:
Jon C. Helton

Journal:
Proc. Amer. Math. Soc. **39** (1973), 155-162

MSC:
Primary 26A39; Secondary 46J99

MathSciNet review:
0316643

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Abstract: Functions are from to , where represents the real numbers and represents a commutative Banach algebra with identity element. The function on only if exists and is not zero and there exists a subdivision of and a number such that if is a refinement of , then exists and . If on , then each of the following consists of two equivalent statements: A. (1) on , and (2) exists. B. (1) on and , and (2) . Further, if on , each of and exist for and has bounded variation on , then each of the following consists of two equivalent statements: C. (1) on , and (2) exists. D. (1) on and , and (2) .

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DOI:
https://doi.org/10.1090/S0002-9939-1973-0316643-3

Keywords:
Sum integral,
product integral,
subdivision-refinement integral,
interval function,
interdependency,
exponential,
commutative Banach algebra

Article copyright:
© Copyright 1973
American Mathematical Society