Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-27T00:02:11.672Z Has data issue: false hasContentIssue false
  • English
  • Français

Bilinear integration with positive vector measures

Part of: Set functions, measures and integrals with values in abstract spaces Normed linear spaces and Banach spaces; Banach lattices Measures, integration, derivative, holomorphy Special classes of linear operators

Published online by Cambridge University Press:  09 April 2009

Brian Jefferies  and
Paul Rothnie
Paul Rothnie
Affiliation:
School of Mathematics The University of New South WalesNSW 2052Australia e-mail: brianj@maths.unsw.edu.au, rothniep@maths.unsw.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The integration of vector (and operator) valued functions with respect to vector (and operator) valued measures can be simplified by assuming that the measures involved take values in the positive elements of a Banach lattice.

MSC classification

Secondary: 28B05: Vector-valued set functions, measures and integrals 46G10: Vector-valued measures and integration 46B42: Banach lattices 47B65: Positive operators and order-bounded operators
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 75 , Issue 2 , October 2003 , pp. 279 - 294
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Bartle, R., ‘A general bilinear vector integral’, Studia Math. 15 (1956), 337–351.CrossRefGoogle Scholar
[2]Diestel, J. and Uhl, J. J. Jr, Vector measures, Math Surveys 15 (Amer. Math. Soc., Providence, 1977).CrossRefGoogle Scholar
[3]Dobrakov, I., ‘On integration in Banach spaces, I’, Czech. Math. J. 20 (1970), 511–536.CrossRefGoogle Scholar
[4]Dobrakov, I., ‘On integration in Banach spaces, II’, Czech. Math. J. 20 (1970), 680–695.CrossRefGoogle Scholar
[5]Dobrakov, I., ‘On representation of linear operators on C o(T, X)’, Czech. Math. J. 21 (1971), 13–30.CrossRefGoogle Scholar
[6]Folland, G., Real analysis modern techniques and their applications’, (John Wiley & Sons, New York, 1984).Google Scholar
[7]Gowurin, M., ‘Über die Stieltjessche Integration abstrakter Funktionen’, Fund. Math. 27 (1936), 255–268.CrossRefGoogle Scholar
[8]Jefferies, B. and Okada, S., ‘Dominated semigroups of operators and evolution processes’, Hokkaido Math. J., to appear.Google Scholar
[9]Jefferies, B. and Okada, S., ‘Bilinear integration in tensor products’, Rocky Mountain J. Math. 28 (1998), 517–545.CrossRefGoogle Scholar
[10]Jefferies, B. R. F., Evolution processes and the Feynman-Kac formula (Kluwer, Dordrecht, 1996).CrossRefGoogle Scholar
[11]Kluvánek, I. and Knowles, G., Vector measures and control systems (North Holland, Amsterdam, 1976).Google Scholar
[12]de La Barriere, R. Pallu, ‘Integration of vector functions with respect to vector measures’, Studia Univ. Babes Bolyai, Mathematica, Vol. XLIII, No. 2, 06 1998.Google Scholar
[13]Reed, M. and Simon, B., Methods of modern Mathematical Physics I–II (Academic Press, New York, 1973).Google Scholar
[14]Rothnie, P., Bilinear integration and the Feynman-Kac formula (Ph. D. Thesis, University of New South Wales, 2000).Google Scholar
[15]Schaefer, H. H., Banach lattices and positive operators, Grundlehren Math. Wiss. 215 (Springer, Berlin, 1974).CrossRefGoogle Scholar
[16]Schmidt, K., Jordan decompositions of generalized vector measures, Pitman Research Notes in Math. 214 (Longman, 1989).Google Scholar