Hostname: page-component-77c89778f8-gq7q9 Total loading time: 0 Render date: 2024-07-21T19:25:31.007Z Has data issue: false hasContentIssue false
  • English
  • Français

Boolean algebras of projections and resolutions of the identity of scalar-type spectral operators

Published online by Cambridge University Press:  20 January 2009

B. de Pagter  and
W. J. Ricker
B. de Pagter
Affiliation:
Faculty of Technical Mathematics & Informatics, Department of Pure Mathematics, Delft University of Technology, Mekelweg 4, 2628CD Delft, The Netherlands
W. J. Ricker
Affiliation:
School of Mathematics, University of New South Wales, Sydney, NSW, 2052, 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.

Let Μ be a Bade complete (or σ-complete) Boolean algebra of projections in a Banach space X. This paper is concerned with the following questions: When is Μ equal to the resolution of the identity (or the strong operator closure of the resolution of the identity) of some scalar-type spectral operator T (with σ(T) ⊆ ℝ) in X? It is shown that if X is separable, then Μ always coincides with such a resolution of the identity. For certain restrictions on Μ some positive results are established in non-separable spaces X. An example is given for which Μ is neither a resolution of the identity nor the strong operator closure of a resolution of the identity.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 40 , Issue 3 , October 1997 , pp. 425 - 435
Copyright
Copyright © Edinburgh Mathematical Society 1997

References

REFERENCES

1.Bade, W. G., On Boolean algebras of projections and algebras of operators, Trans. Amer. Math Soc. 80 (1955), 345–359.CrossRefGoogle Scholar
2.Dixmier, J., Von Neumann algebras (North Holland Series, No. 27, Amsterdam, 1981).Google Scholar
3.Dodds, P. G. and Ricker, W. J., Spectral measures and the Bade reflexivity theorem, J. Funct. Anal. 61 (1985), 136–163.CrossRefGoogle Scholar
4.Dunford, N., Spectral theory, II. Resolutions of the identity, Pacific J. Math. 2 (1952), 559–614.CrossRefGoogle Scholar
5.Dunford, N., Spectral operators, Pacific J. Math. 4 (1954), 321–354.CrossRefGoogle Scholar
6.Dunford, N. and Schwartz, J. T., Linear operators I, 2nd Printing (Wiley-Interscience, New York, 1964).Google Scholar
7.Dunford, N. and Schwartz, J. T., Linear operators III: spectral operators, (Wiley-Interscience, New York, 1972).Google Scholar
8.Kantorovitz, S., On the characterization of spectral operators, Trans. Amer. Math. Soc. 111 (1964), 152–181.CrossRefGoogle Scholar
9.Ricker, W. J., Spectral operators of scalar-type in Grothendieck spaces with the Dunford-Pettis property, Bull. London Math. Soc. 17 (1985), 268–270.CrossRefGoogle Scholar
10.Ricker, W. J., Separability of the L1 -space of a vector measure, Glasgow Math. J. 34 (1992), 1–9.CrossRefGoogle Scholar
11.Ricker, W. J., Well bounded operators of type (B) in H.I. spaces, Acta Sci. Math. (Szeged) 59 (1994), 475–488.Google Scholar
12.Royden, H. L., Real analysis, 2nd Edition (Macmillan Co., London, 1970).Google Scholar