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

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

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) .

**[1]**W. P. Davis and J. A. Chatfield,*Concerning product integrals and exponentials*, Proc. Amer. Math. Soc.**25**(1970), 743–747. MR**0267068**, https://doi.org/10.1090/S0002-9939-1970-0267068-8**[2]**Burrell W. Helton,*Integral equations and product integrals*, Pacific J. Math.**16**(1966), 297–322. MR**0188731****[3]**Burrell W. Helton,*A product integral representation for a Gronwall inequality*, Proc. Amer. Math. Soc.**23**(1969), 493–500. MR**0248310**, https://doi.org/10.1090/S0002-9939-1969-0248310-8**[4]**Jon C. Helton,*Some interdependencies of sum and product integrals*, Proc. Amer. Math. Soc.**37**(1973), 201–206. MR**0308340**, https://doi.org/10.1090/S0002-9939-1973-0308340-5**[5]**-,*Product integrals, bounds and inverses*, Texas J. Sci. (to appear).**[6]**J. S. MacNerney,*Integral equations and semigroups*, Illinois J. Math.**7**(1963), 148–173. MR**0144179****[7]**P. R. Masani,*Multiplicative Riemann integration in normed rings*, Trans. Amer. Math. Soc.**61**(1947), 147–192. MR**0018719**, https://doi.org/10.1090/S0002-9947-1947-0018719-6

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Additional Information

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