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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 $ R \times R$ to $ X$, where $ R$ represents the real numbers and $ X$ represents a commutative Banach algebra with identity element. The function $ G \in O{C^ \circ }$ on $ [a,b]$ only if $ {}_a{\prod ^b}(1 + G)$ exists and is not zero and there exists a subdivision $ D$ of $ [a,b]$ and a number $ B$ such that if $ J$ is a refinement of $ D$, then $ {[\prod\nolimits_J {(1 + G)} ]^{ - 1}}$ exists and $ \vert{[\prod\nolimits_J {(1 + G)} ]^{ - 1}}\vert < B$. If $ \vert G\vert < 1$ on $ [a,b]$, then each of the following consists of two equivalent statements: A. (1) $ G \in O{C^ \circ }$ on $ [a,b]$, and (2) $ \int_a^b {\ln (1 + G)} $ exists. B. (1) $ G \in O{C^ \circ }$ on $ [a,b]$ and $ \int_a^b {\vert 1 + G - \prod {(1 + G)} \vert = 0} $, and (2) $ \int_a^b {\vert\ln (1 + G) - \int {\ln (1 + G)\vert} = 0} $. Further, if $ \beta > 0,\vert G\vert < 1 - \beta $ on $ [a,b]$, each of $ G(p,{p^ + }),G({p^ - },p),G({p^ + },{p^ + })$ and $ G({p^ - },{p^ - })$ exist for $ p \in [a,b],\int_a^b {\vert{G^2} - \int {{G^2}\vert} } = 0$ and $ {G^2}$ has bounded variation on $ [a,b]$, then each of the following consists of two equivalent statements: C. (1) $ G \in O{C^ \circ }$ on $ [a,b]$, and (2) $ \int_a^b G $ exists. D. (1) $ G \in 0{C^ \circ }$ on $ [a,b]$ and $ \int_a^b {\vert 1 + G} - \prod {(1 + G)} \vert = 0$, and (2) $ \int_a^b {\vert G - \int {G\vert} = 0} $.


References [Enhancements On Off] (What's this?)

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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

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