Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-28T17:28:19.206Z Has data issue: false hasContentIssue false
  • English
  • Français

Scalar operators and integration

Part of: Special classes of linear operators Measures, integration, derivative, holomorphy Abstract harmonic analysis

Published online by Cambridge University Press:  09 April 2009

Igor Kluvánek
Igor Kluvánek
Affiliation:
Centre for Mathematical Analysis, Australian National UniversityP. O. Box 4, Canberra, ACT 2600, 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.

The notion of a scalar operator on a Banach space, in the sense of N. Dunford, is widened so as to cover those operators which can be approximated in the operator norm by linear combinations of disjoint values of an additive and multiplicative operator valued set function, P, on an algebra of sets in a space Ω such that P(Ω) = I, subject to some conditions guaranteeing that this definition is unambiguous. An operator T turns out to be scalar in this sense, if and only if, there exists a (not necessarily bounded) Boolean algebra of bounded projections such that the Banach algebra of operators it generates is semisimple and contains T.

MSC classification

Secondary: 46G10: Vector-valued measures and integration 47B40: Spectral operators, decomposable operators, well-bounded operators, etc. 43A22: Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 45 , Issue 3 , December 1988 , pp. 401 - 420
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Colojoarâ, I. and Foiaş, C., Theory of generalized spectral operators, (Gordon and Breath, New York, 1968).Google Scholar
[2]Dodds, P. G. and Ricker, W., ‘Spectral measures and the Bade reflexivity theory’, J. Funct. Anal. 61 (1985), 136–163.CrossRefGoogle Scholar
[3]Dowson, H. R., Spectral theory of linear operators, (London Math. Soc. Monographs, No. 12, Academic Press, New York, 1968).Google Scholar
[4]Dunford, N. and Schwartz, J. T., Linear operators, Part III: Spectral operators, (WileyInterscience, New York, 1971).Google Scholar
[5]Caudry, G. I. and Ricker, W., ‘Spectral properties of Lp translations’, J. Operator Theory 14 (1985), 87–111.Google Scholar
[6]Gillespie, T. A., ‘A spectral theorem for Lp translations’, J. London Math. Soc. (2) 11 (1975), 449–508.Google Scholar
[7]Hirschman, I. I., ‘On multiplier transformations’, Duke Math. J. 26 (1959), 221–242.CrossRefGoogle Scholar
[8]Kluvánek, I., The extension and closure of vector measures, Vector and operator-valued measures and applications, Proc. Sympos., Alta, Utah, 1972, pp. 175–190, (Academic Press, New York, 1973).Google Scholar
[9]Loomis, L. H., An introduction to abstract harmonic analysis, (D. Van Nostrand Co., New York, 1953).Google Scholar
[10]Ricker, W., ‘Extended spectral operators’, J. Operator Theory 9 (1983), 269–296.Google Scholar
[11]Ringrose, J. R., ‘On well-bounded operators’, J. Austral. Math. Soc. 1 (1960), 334–343.CrossRefGoogle Scholar
[12]Smart, D. R., ‘Conditionally convergent spectral expansions’, J. Austral. Math. Soc. 1 (1960), 319–333.CrossRefGoogle Scholar