A derivative in the setting of operator-valued measures

Authors:
J. R. Edwards and S. G. Wayment

Journal:
Proc. Amer. Math. Soc. **38** (1973), 523-531

MSC:
Primary 28A45; Secondary 28A15, 46G10

DOI:
https://doi.org/10.1090/S0002-9939-1973-0311870-3

MathSciNet review:
0311870

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Abstract: There has been considerable recent development concerning integration of vector-valued functions with respect to operator-valued measures, including various Radon-Nikodym type theorems. This paper is concerned with obtaining an analytic definition of derivative which will act as an inverse operator for these integration processes. A derivative is defined and, with additional conditions imposed on the measure, the two fundamental theorems of calculus are proved, i.e. that differentiation and integration are indeed inverse processes. Several examples are discussed to point out the shortcomings of this definition of derivative. Since this paper was written, some partial work has been done by D. H. Tucker and S. G. Wayment to adjust the definition to remove some of these defects. However, they were not then able to establish certain desirable theorems. For further details, the reader is referred to the Proc. Sympos. on Vector and Operator Valued Measures and Applications (Snowbird, Utah, 1972).

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1973-0311870-3

Keywords:
Linear topological,
Hausdorff space,
locally compact Hausdorff space,
bounded semivariation,
weakly differentiable,
strongly differentiable,
locally nondispersive,
Fundamental Theorem of Calculus

Article copyright:
© Copyright 1973
American Mathematical Society