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Integration with respect to the product of a measure and a closable set function

Published online by Cambridge University Press:  22 January 2016

Brian Jefferies
Brian Jefferies*
Affiliation:
Department of Mathematics, The University of Wollongong, P.O. Box 1144 Wollongong, N.S.W. 2500, Australia
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The notion of a set function being closable with respect to a family of measures was introduced in [1] and applied there to the integration of Feynman-Kac functionals. In [2], it is shown how the interchange of integrals for the product of an operator valued measure and a scalar measure can be used to solve operator equations associated with the perturbation of semigroups. The purpose of this article is to develop the analogous machinery for closable set functions with the view of applying it to Schrödinger’s equation.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 115 , September 1989 , pp. 47 - 62
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1989

Footnotes

*

Research supported by a Queen Elizabeth II Fellowship.

References

[1] Jefferies, B., Integration with respect to closable set functions, J. Funct. Anal., 66 (1986), 381–405.Google Scholar
[2] Kluvánek, I., Operator valued measures and perturbations of semigroups, Arch. Rat. Mech. Anal., 81 (1983), 161–180.Google Scholar
[3] Kluvánek, I., Integration and the Feynman-Kac formula Studia Mathematica, 86 (1987), 35–57.Google Scholar
[4] Kluvánek, I. and Knowles, J., Vector Measures and Control Systems, North Holland, Amsterdam, 1976.Google Scholar
[5] Schaefer, H. H., Topological Vector Spaces, Springer-Verlag, New York, Heidelberg, Berlin 1971.Google Scholar
[6] Simon, B., Essential self-adjointness of Schrödinger operators with positive potentials, Math. Ann., 201 (1973), 211–220.Google Scholar