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

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

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 value function space integral and a related integral equation*, J. Math. Mech.**18**(1968), 517-552. MR**38**#4643. MR**0236347 (38:4643)****[2]**-,*An integral equation related to the Schroedinger equation with an application to integration in function space*, Bochner Memorial Volume (to appear).**[3]**E. Hille,*Analytic function theory*. Vol. II, Introduction to Higher Math., Ginn, Boston, Mass., 1962. MR**34**#1490. MR**0201608 (34:1490)****[4]**E. Hille and R. S. Phillips,*Functional analysis and semi-groups*, rev. ed., Amer. Math. Soc. Colloq. Publ., vol. 31, Amer. Math. Soc, Providence, R.I., 1957. MR**19**, 664. MR**0089373 (19:664d)****[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 (40:7882)****[6]**E. Nelson,*Feynman integrals and the Schrödinger equation*, J. Mathematical Phys.**5**(1964), 332-343. MR**28**#4397. MR**0161189 (28:4397)****[7]**D. L. Skoug and G. W. Johnson,*Operator-valued Feynman integrals of finite-dimensional functionals*, Pacific J. Math.**34**(1970), 415-426. MR**0268728 (42:3625)**

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Additional Information

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