Operator-valued Feynman integrals of certain finite-dimensional functionals

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

Journal:
Proc. Amer. Math. Soc. **24** (1970), 774-780

MSC:
Primary 47.70; Secondary 28.00

DOI:
https://doi.org/10.1090/S0002-9939-1970-0254675-1

MathSciNet review:
0254675

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Abstract | References | Similar Articles | Additional Information

Abstract: Let denote the space of continuous functions on such that . Let where . Recently, Cameron and Storvick defined an operator-valued ``Feynman Integral.'' In their setting, we give a strong existence theorem as well as an explicit formula for the ``Feynman Integral'' of functionals as above under weak restrictions on the 's. We also give necessary and sufficient conditions for the operator to be invertible and an explicit formula for the inverse.

**[1]**R. H. Cameron,*The Ilstow and Feynman integrals*, J. Analyse Math.**10**(1962/1963), 287–361. MR**0150845**, https://doi.org/10.1007/BF02790311**[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**0236347****[3]**Nelson Dunford and Jacob T. Schwartz,*Linear Operators. I. General Theory*, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. MR**0117523****[4]**Paul R. Halmos,*A Hilbert space problem book*, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR**0208368****[5]**D. L. Skoug,*Generalized Ilstow and Feynman integrals*, Pacific J. Math.**26**(1968), 171–192. MR**0230870**

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DOI:
https://doi.org/10.1090/S0002-9939-1970-0254675-1

Keywords:
Feynman integral,
Wiener integral,
convolution operator,
Fourier transform,
operator valued Feynman integral

Article copyright:
© Copyright 1970
American Mathematical Society