Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T14:59:01.633Z Has data issue: false hasContentIssue false
  • English
  • Français

The spectral theorem for well-bounded operators

Part of: Special classes of linear operators Topological linear spaces and related structures Topological algebras, normed rings and algebras, Banach algebras General theory of linear operators

Published online by Cambridge University Press:  09 April 2009

Ian Doust  and
Qiu Bozhou
Ian Doust
Affiliation:
School of Mathematics, University of New South Wales, Kensington, NSW, 2033, Australia
Qiu Bozhou
Affiliation:
Department of Applied Mathematics, Tongji University, Shanghai, People's Republic of, China
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.

Well-bounded operators are those which possess a bounded functional calculus for the absolutely continuous functions on some compact interval. Depending on the weak compactness of this functional calculus, one obtains one of two types of spectral theorem for these operators. A method is given which enables one to obtain both spectral theorems by simply changing the topology used. Even for the case of well-bounded operators of type (B), the proof given is more elementary than that previously in the literature.

MSC classification

Secondary: 47B40: Spectral operators, decomposable operators, well-bounded operators, etc. 46A50: Compactness in topological linear spaces; angelic spaces, etc. 46H30: Functional calculus in topological algebras 47A60: Functional calculus
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 54 , Issue 3 , June 1993 , pp. 334 - 351
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Barry, J. Y., ‘On the convergence of ordered sets of projections’, Proc. Amer. Math. Soc. 5 (1954), 313–314.Google Scholar
[2]Berkson, E., ‘A Fourier analysis theory of abstract spectral decompositions’, in Surveys of some recent results in operator theory, Vol. I (eds. Conway, J. B. and Morrel, B. B.), Pitman Res. Notes Math. Ser. 171 (Longman Sci. Tech., Harlow, 1988) pp. 95–132.Google Scholar
[3]Berkson, E. and Dowson, H. R., ‘On uniquely decomposable well-bounded operators’, Proc. London Math. Soc. (3) 22 (1971), 339–358.CrossRefGoogle Scholar
[4]Berkson, E. and Gillespie, T. A., ‘AC functions on the circle and spectral families’, J. Operator Theory 13 (1985), 33–47.Google Scholar
[5]Berkson, E. and Gillespie, T. A., ‘Stečkin's theorem, transference, and spectral decompositions’, J. Funct. Anal. 70 (1987), 140–170.Google Scholar
[6]Dowson, H. R., Spectral theory of linear operators, London Math. Soc. Monographs 12 (Academic Press, London, 1978).Google Scholar
[7]Halmos, P. R., Measure theory (Van Nostrand, Princeton, 1950).CrossRefGoogle Scholar
[8]Ringrose, J. R., ‘On well-bounded operators’, J. Austral Math. Soc. (Series A) 1 (1960), 334–343.Google Scholar
[9]Ringrose, J. R., ‘On well-bounded operators II’, Proc. London Math. Soc. (3) 13 (1963), 613–638.CrossRefGoogle Scholar
[10]Sills, W. H., ‘On absolutely continuous functions and the well-bounded operator’, Pacific J. Math. 17 (1966), 349–366.CrossRefGoogle Scholar
[11]Smart, D. R., ‘Conditionally convergent spectral expansions’, J. Austral Math. Soc. (Series A) 1 (1960), 319–333.Google Scholar
[12]Spain, P. G., ‘On well-bounded operators of type (B)’, Proc. Edinburgh Math. Soc. (2) 18 (1972), 35–48.CrossRefGoogle Scholar