Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] N. Dunford and J. T. Schwartz, Linear operators I: General theory, Pure and Appl. Math., vol. 7, Interscience, New York, 1958. MR 22 #8302. MR 0117523 (22:8302)
  • [2] R. J. Easton and D. H. Tucker, A generalized Lebesgue-type integral, Math. Ann. 181 (1969), 311-324. MR 1513279
  • [3] 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 43 #7581. MR 0281867 (43:7581)
  • [4] R. K. Goodrich, A Riesz representation theorem in the setting of locally convex spaces, Trans. Amer. Math. Soc. 131 (1968), 246-258. MR 36 #5731. MR 0222681 (36:5731)
  • [5] H. B. Maynard, A Radon-Nikodym theorem for finitely additive bounded measures, Pacific J. Math. (to appear). MR 557942 (80k:28007)
  • [6] D. 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 33 #7865. MR 0199722 (33:7865)
  • [7] D. H. Tucker and S. G. Wayment, Absolute continuity and the Radon-Nikodym theorem, J. Reine Angew. Math. 244 (1970), 1-19. MR 42 #7859. MR 0272978 (42:7859)
  • [8] A. H. Shuchat, Integral representation theorems in topological vector spaces, Trans. Amer. Math. Soc. 172 (1972), 373-397. MR 0312264 (47:826)
  • [9] S. G. Wayment, Absolute continuity and the Radon theorem, Ph.D. Thesis, University of Utah, Salt Lake City, Utah, 1968.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 28A45, 28A15, 46G10

Retrieve articles in all journals with MSC: 28A45, 28A15, 46G10


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

American Mathematical Society