A derivative in the setting of operator-valued measures

Edwards, J. R.; Wayment, S. G.

doi:10.1090/S0002-9939-1973-0311870-3

A derivative in the setting of operator-valued measures
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by J. R. Edwards and S. G. Wayment PDF

Proc. Amer. Math. Soc. 38 (1973), 523-531 Request permission

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).

References

Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
R. J. Easton and D. H. Tucker, A generalized lebesgue-type integral, Math. Ann. 181 (1969), no. 4, 311–324. MR 1513279, DOI 10.1007/BF01350670
J. R. Edwards and S. G. Wayment, Integral representations for continuous linear operators in the setting of convex topological vector spaces, Trans. Amer. Math. Soc. 157 (1971), 329–345. MR 281867, DOI 10.1090/S0002-9947-1971-0281867-3
Robert K. Goodrich, A Riesz representation theorem in the setting of locally convex spaces, Trans. Amer. Math. Soc. 131 (1968), 246–258. MR 222681, DOI 10.1090/S0002-9947-1968-0222681-4
Hugh B. Maynard, A Radon-Nikodým theorem for finitely additive bounded measures, Pacific J. Math. 83 (1979), no. 2, 401–413. MR 557942
Don H. Tucker, A representation theorem for a continuous linear transformation on a space of continuous functions, Proc. Amer. Math. Soc. 16 (1965), 946–953. MR 199722, DOI 10.1090/S0002-9939-1965-0199722-9
D. H. Tucker and S. G. Wayment, Absolute continuity and the Radon-Nikodym theorem, J. Reine Angew. Math. 244 (1970), 1–19. MR 272978, DOI 10.1515/crll.1970.244.1
Alan H. Shuchat, Integral representation theorems in topological vector spaces, Trans. Amer. Math. Soc. 172 (1972), 373–397. MR 312264, DOI 10.1090/S0002-9947-1972-0312264-0

Absolute continuity and the Radon theorem