An operator valued function space integral: A sequel to Cameron and Storvick's paper

Authors:
G. W. Johnson and D. L. Skoug

Journal:
Proc. Amer. Math. Soc. **27** (1971), 514-518

MSC:
Primary 28.46

MathSciNet review:
0272974

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Abstract: Recently Cameron and Storvick introduced and studied an operator valued function space integral related to the Feynman integral. The main theorems of their study establish the existence of the function space integral as a weak operator limit of operators defined at the first stage by finite-dimensional integrals. This paper provides a substantial strengthening of their existence theorem giving the function space integrals as strong operator limits rather than as weak operator limits.

**[1]**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**0236347****[2]**-,*An integral equation related to the Schroedinger equation with an application to integration in function space*, Bochner Memorial Volume (to appear).**[3]**Einar Hille,*Analytic function theory. Vol. II*, Introductions to Higher Mathematics, Ginn and Co., Boston, Mass.-New York-Toronto, Ont., 1962. MR**0201608****[4]**Einar Hille and Ralph S. Phillips,*Functional analysis and semi-groups*, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957. rev. ed. MR**0089373****[5]**G. W. Johnson and D. L. Skoug,*Operator-valued Feynman integrals of certain finite-dimensional functionals*, Proc. Amer. Math. Soc.**24**(1970), 774–780. MR**0254675**, 10.1090/S0002-9939-1970-0254675-1**[6]**Edward Nelson,*Feynman integrals and the Schrödinger equation*, J. Mathematical Phys.**5**(1964), 332–343. MR**0161189****[7]**G. W. Johnson and D. L. Skoug,*Operator-valued Feynman integrals of finite-dimensional functionals.*, Pacific J. Math.**34**(1970), 415–425. MR**0268728**

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DOI:
https://doi.org/10.1090/S0002-9939-1971-0272974-5

Keywords:
Feynman integral,
Wiener integral,
operator valued function space integral,
Poisson representation,
Bochner integral

Article copyright:
© Copyright 1971
American Mathematical Society