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A function space integral for a Banach space of functionals on Wiener space


Authors: G. W. Johnson and D. L. Skoug
Journal: Proc. Amer. Math. Soc. 43 (1974), 141-148
MSC: Primary 28A40; Secondary 46G10
DOI: https://doi.org/10.1090/S0002-9939-1974-0340536-X
MathSciNet review: 0340536
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Abstract: In an earlier paper the authors established the existence of Cameron and Storvick's function space integral $ {J_q}(F)$ for a class of finite-dimensional functionals $ F$. Here we consider a space $ A$ of not necessarily finite-dimensional functionals generated by the earlier functionals. We show that $ A$ is a Banach space and recognize $ A$ as the direct sum of more familiar Banach spaces. We also show that the function space integral $ J_q^{{\text{an}}}(F)$ exists for $ F \in A$. In contrast we give an example of an $ {F_0} \in A$ such that $ J_q^{{\text{seq}}}({F_0})$ does not exist.


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

  • [1] John A. Beekman and Ralph A. Kallman, Gaussian Markov expectations and related integral equations, Pacific J. Math. 37 (1971), 303-318. MR 0308353 (46:7467)
  • [2] R. H. Cameron and D. A. Storvick, An operator valued function space integral and a related integral equation, J. Math. Mech. 18 (1968), 517-552. MR 38 #4643. MR 0236347 (38:4643)
  • [3] G. W. Johnson and D. L. Skoug, Operator-valued Feynman integrals of finite-dimensional functionals, Pacific J. Math. 34 (1970), 415-425. MR 42 #3625. MR 0268728 (42:3625)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1974-0340536-X
Keywords: Wiener integral, operator valued function space integral, Feynman integral, Banach space
Article copyright: © Copyright 1974 American Mathematical Society

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