Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-29T18:40:01.877Z Has data issue: false hasContentIssue false
  • English
  • Français

Integration maps and local equicontinuity of spectral measures

Part of: Special classes of linear operators

Published online by Cambridge University Press:  09 April 2009

S. Okada  and
W. J. Ricker
W. J. Ricker
Affiliation:
School of Mathematics University of New South Wales Sydney, NSW 2052Australia e-mail: okada@maths.unsw.edu.au e-mail: werner@maths.unsw.edu.au
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.

One of the useful features of spectral measures which happen to be equicontinuous is that their associated integration maps are bicontinuous isomorphisms of the corresponding L1-space onto their ranges. It is shown here that equicontinuity is not necessary for this to be the case; a somewhat weaker property suffices. This is of some interest in practice since there are many natural examples of spectral measures which fail to be equiconontinuous.

MSC classification

Secondary: 47B15: Hermitian and normal operators (spectral measures, functional calculus, etc.) 47B40: Spectral operators, decomposable operators, well-bounded operators, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 69 , Issue 2 , October 2000 , pp. 176 - 186
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Dodds, P. G. and de Pagter, B., ‘Orthomorphisms and Boolean algebras of projections’, Math. Z. 187 (1984), 361–381.CrossRefGoogle Scholar
[2]Dodds, P. G. and de Pagter, B., ‘Algebras of unbounded scalar-type spectral operators’, Pacific. J. Math. 130 (1987), 41–74.CrossRefGoogle Scholar
[3]Dodds, P. G., de Pagter, B. and Ricker, W. J., ‘Reflexivity and order properties of scalar-type spectral operators in locally convex spaces’, Trans. Amer. Math. Soc. 293 (1986), 355–380.CrossRefGoogle Scholar
[4]Dodds, P. G. and Ricker, W. J., ‘Spectral measures and the Bade reflexivity theorem’, J. Funct. Anal. 61 (1985), 136–163.CrossRefGoogle Scholar
[5]Dunford, N. and Schwartz, J. T., Linear operators III: Spectral operators (Wiley-Interscience, New York, 1971).Google Scholar
[6]Ionescu-Tulcea, C., ‘Spectral operators on locally convex spaces’, Bull. Amer. Math. Soc. 67 (1961), 125–128.CrossRefGoogle Scholar
[7]Kluvánek, I. and Knowles, G., Vector measures and control systems (North Holland, Amsterdam, 1976).Google Scholar
[8]Lewis, D. R., ‘Integration with respect to vector measures’, Pacific J. Math. 33 (1970), 157–165.CrossRefGoogle Scholar
[9]Okada, S. and Ricker, W. J., ‘Spectral measures which fail to be equicontinuous’, Period. Math. Hungar. 28 (1994), 55–61.CrossRefGoogle Scholar
[10]Okada, S. and Ricker, W. J., ‘Vector measures and integration in non-complete spaces’, Arch. Math. (Basel) 63 (1994), 334–353.CrossRefGoogle Scholar
[11]Okada, S. and Ricker, W. J., ‘The range of the integration map of a vector measure’, Arch. Math. (Basel) 64 (1995), 512–522.CrossRefGoogle Scholar
[12]Okada, S. and Ricker, W. J., ‘Continuous extensions of spectral measures’, Colloq. Math. 71 (1996), 115–132.CrossRefGoogle Scholar
[13]Okada, S. and Ricker, W. J., ‘Spectral measures and automatic continuity’, Bull. Belgian Math. Soc. 3 (1996), 267–279.Google Scholar
[14]Okada, S. and Ricker, W. J., ‘Boolean algebras of projections and ranges of spectral measures’, Dissertationes Math. 365 (1997), 33pp.Google Scholar
[15]Ricker, W. J., ‘Boolean algebras of projections and spectral measures in dual spaces’, Operator Theory: Adv. Appl. 43 (1990), 289–300.Google Scholar
[16]Ricker, W. J., ‘Completeness of the L1-space of closed vector measures’, Proc. Edinburgh Math. Soc. 33 (1990), 71–78.CrossRefGoogle Scholar
[17]Ricker, W. J., ‘The sequential closedness of σ-complete Boolean algebras of projections’, J. Math. Anal. Appl. 208 (1997), 364–371.CrossRefGoogle Scholar
[18]Ricker, W. J. and Schaefer, H. H., ‘The uniformly closed algebra generated by a complete Boolean algebra of projections’, Math. Z. 201 (1989), 429–439.CrossRefGoogle Scholar
[19]Schaefer, H. H., ‘Spectral measures in locally convex algebras’, Acta Math. 107 (1962), 125–173.CrossRefGoogle Scholar
[20]Shuchat, A., ‘Vector measures and sclar operators in locally convex spaces’, Michigan Math. J. 24 (1977), 303–310.CrossRefGoogle Scholar
[21]Walsh, B., ‘Structure of spectral measures on locally convex spaces’, Trans. Amer. Math. Soc. 120 (1965), 295–326.CrossRefGoogle Scholar