Product integrals and exponentials in commutative Banach algebras
Author:
Jon C. Helton
Journal:
Proc. Amer. Math. Soc. 39 (1973), 155162
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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 B. W. Helton, Integral equations and product integrals, Pacific J. Math. 16 (1966), 297322. MR 32 #6167. MR 0188731 (32:6167)
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 , A product integral representation for a Gronwall inequality, Proc. Amer. Math. Soc. 23 (1969), 493500. MR 40 #1562. MR 0248310 (40:1562)
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 J. C. Helton, Some interdependencies of sum and product integrals, Proc. Amer. Math. Soc. 37 (1973), 201206. MR 0308340 (46:7454)
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 , Product integrals, bounds and inverses, Texas J. Sci. (to appear).
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 J. S. MacNerney, Integral equations and semigroups, Illinois J. Math. 7 (1963), 148173. MR 26 #1726. MR 0144179 (26:1726)
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 P. R. Masani, Multiplicative Riemann integration in normed rings, Trans. Amer. Math. Soc. 61 (1947), 147192. MR 8, 321. MR 0018719 (8:321c)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197303166433
PII:
S 00029939(1973)03166433
Keywords:
Sum integral,
product integral,
subdivisionrefinement integral,
interval function,
interdependency,
exponential,
commutative Banach algebra
Article copyright:
© Copyright 1973 American Mathematical Society
