Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-24T13:55:28.796Z Has data issue: false hasContentIssue false
  • English
  • Français

A concrete realization of the dual space of L1-spaces of certain vector and operator-valued measures

Part of: Set functions, measures and integrals with values in abstract spaces Measures, integration, derivative, holomorphy Linear function spaces and their duals

Published online by Cambridge University Press:  09 April 2009

Werner Ricker
Werner Ricker
Affiliation:
School of Mathematics and Physics Macquarie University North Ryde, NSW 2113, Australia
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.

For a given vector measure n, an important problem, but in practice a difficult one, is to give a concrete description of the dual space of L1(n). In this note such a description is presented for an important class of measures n, namely the spectral measures (in the sense of N. Dunford) and certain other vector and operator-valued measures that they naturally induce. The basic idea is to represent the L1-spaces of such measures as a more familiar space whose dual space is known.

MSC classification

Secondary: 28B05: Vector-valued set functions, measures and integrals 46G10: Vector-valued measures and integration 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 42 , Issue 2 , April 1987 , pp. 265 - 279
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Dodds, P. G., de Pagter, B. and Ricker, W., ‘Reflexivity and order properties of scalar-type spectral operators in locally convex spaces’, Trans. Amer. Math. Soc. 293 (1986), 355–380.CrossRefGoogle Scholar
[2]Dodds, P. G. and Ricker, W., ‘Spectral measures and the Bade reflexivity theorem’, J. Funct. Anal. 61 (1985), 136–163.CrossRefGoogle Scholar
[3]Dunford, N. and Schwartz, J., Linear operators I: General theory (Wiley-Interscience, New York, 1964).Google Scholar
[4]Dunford, N. and Schwartz, J., Linear operators III: Spectral operators (Wiley-Interscience, New York, 1971).Google Scholar
[5]Gillespie, T. A., ‘Boolean algebras of projections and reflexive algebras of operators’, Proc. London Math. Soc. 37 (1978), 56–74.CrossRefGoogle Scholar
[6]Hoffman-Jørgensen, J., ‘Vector measures’, Math. Scand. 28 (1971), 5–32.CrossRefGoogle Scholar
[7]Kluvanek, I., ‘The range of a vector-valued measure’, Math. Systems Theory 7 (1973), 44–54.CrossRefGoogle Scholar
[8]Kluvanek, I., ‘Conical measures and vector measures’, Ann. Inst. Fourier (Grenoble) 27 (1), (1977), 83–105.CrossRefGoogle Scholar
[9]Kluvanek, I. and Knowles, G., Vector measures and control systems (North-Holland, Amsterdam, 1976).Google Scholar
[10]Lewis, D. R., ‘On integrability and summability in vector spaces’, Illinois J. Math. 16 (1972), 294–307.CrossRefGoogle Scholar
[11]Ricker, W., ‘On Boolean algebras of projections and scalar-type spectral operators’, Proc. Amer. Math. Soc. 87 (1983), 73–77.CrossRefGoogle Scholar
[12]Ricker, W., ‘Criteria for closedness of vector measures’, Proc. Amer. Math. Soc. 91 (1984), 75–80.CrossRefGoogle Scholar
[13]Ricker, W., ‘Closed spectral measures in Fréchet spaces’, Internat. J. Math. & Math. Sci. 7 (1984), 15–21.CrossRefGoogle Scholar
[14]Ricker, W., ‘A spectral mapping theorem for scalar-type spectral operators in locally convex spaces’, Integral Equations Operator Theory 8 (1985), 276–288.CrossRefGoogle Scholar
[15]Walsh, B., ‘Structure of spectral measures on locally convex spaces’, Trans. Amer. Math. Soc. 120 (1965), 295–326.CrossRefGoogle Scholar