Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-27T00:07:50.435Z Has data issue: false hasContentIssue false
  • English
  • Français

FUNCTIONAL CALCULUS EXTENSIONS ON DUAL SPACES

Part of: Special classes of linear operators

Published online by Cambridge University Press:  10 March 2009

VENTA TERAUDS
VENTA TERAUDS*
Affiliation:
School of Mathematical and Physical Sciences, University of Newcastle, University Drive, Callaghan NSW 2308, Australia (email: [email protected])
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.

In this note, we show that if a Banach space X has a predual, then every bounded linear operator on X with a continuous functional calculus admits a bounded Borel functional calculus. A consequence of this result is that on such a Banach space, the classes of finitely spectral and prespectral operators coincide. We also apply our theorem to give some sufficient conditions for an operator with an absolutely continuous functional calculus to admit a bounded Borel one.

Keywords

functional calculusfinitely spectral operatorswell-bounded operatorsAC(σ) operators

MSC classification

Secondary: 47B40: Spectral operators, decomposable operators, well-bounded operators, etc.
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 79 , Issue 1 , February 2009 , pp. 71 - 77
Copyright
Copyright © Australian Mathematical Society 2009

References

[1] Ashton, B. and Doust, I., ‘Functions of bounded variation on compact subsets of the plane’, Studia Math. 169 (2005), 163–188.CrossRefGoogle Scholar
[2] Berkson, E. and Dowson, H. R., ‘On uniquely decomposable well-bounded operators’, Proc. London Math. Soc. (3) 22 (1971), 339–358.CrossRefGoogle Scholar
[3] Berkson, E. and Dowson, H. R., ‘Prespectral operators’, Illinois J. Math. 13 (1969), 291–315.CrossRefGoogle Scholar
[4] Doust, I. and deLaubenfels, R., ‘Functional calculus, integral representations, and Banach space geometry’, Quaest. Math. 17(2) (1994), 161–171.CrossRefGoogle Scholar
[5] Doust, I. and Terauds, V., ‘Extensions of an AC(σ) functional calculus’, submitted.Google Scholar
[6] Dowson, H. R., Spectral Theory of Linear Operators (Academic Press, London, 1978).Google Scholar
[7] Dunford, N., ‘Spectral operators’, Pacific J. Math. 4 (1954), 321–354.CrossRefGoogle Scholar
[8] Dunford, N. and Schwartz, J. T., Linear Operators. Part III (Interscience Publishers, John Wiley & Sons, Inc., New York, 1971).Google Scholar
[9] Gillespie, T. A., ‘Spectral measures on spaces not containing ’, Proc. Edinburgh Math. Soc. (2) 24(1) (1981), 41–45.CrossRefGoogle Scholar
[10] Nagy, B., ‘Finitely spectral operators’, Glasgow Math. J. 28(1) (1986), 95–112.CrossRefGoogle Scholar
[11] Sills, W. H., ‘On absolutely continuous functions and the well-bounded operator’, Pacific J. Math. 17 (1966), 349–366.CrossRefGoogle Scholar
[12] Spain, P. G., ‘On scalar-type spectral operators’, Proc. Cambridge Philos. Soc. 69 (1971), 409–410.CrossRefGoogle Scholar
[13] Terauds, V., Extensions of functional calculus for Banach space operators, PhD Thesis, University of New South Wales, 2006.Google Scholar