Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-15T13:44:30.671Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectral properties of elements in different Banach algebras

Published online by Cambridge University Press:  18 May 2009

J. J. Grobler  and
H. Raubenheimer
J. J. Grobler
Affiliation:
Department of Mathematics and Applied Mathematics, Potchefstroom University for Christian Higher Education, Potchefstroom 2520, South Africa
H. Raubenheimer
Affiliation:
Department of Mathematics, University of the Orange Free State, Bloemfontein 9300, South Africa
Rights & Permissions [Opens in a new window]

Extract

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 A be a Banach algebra with unit 1 and let B be a Banach algebra which is a subalgebra of A and which contains 1. In [5]Barnes gave sufficient conditions for B to be inverse closed in A. In this paper we consider single elements and study the question of how the spectrum relative to B of an element in B relates to the spectrum of the element relative to A.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 33 , Issue 1 , January 1991 , pp. 11 - 20
Copyright
Copyright © Glasgow Mathematical Journal Trust 1991

References

REFERENCES

1.Aupetit, B., Inessential elements in Banach algebras, Bull. London Math. Soc. 18 (1986), 493–497.CrossRefGoogle Scholar
2.Arendt, W., On the o-spectrum of regular operators and the spectrum of measures, Math. Z. 178 (1981), 271–287.CrossRefGoogle Scholar
3.Arendt, W. and Hart, D. R., The spectrum of quasi-invertible disjointness preserving operators, J. Functional Analysis, 68 (1986), 149–167.CrossRefGoogle Scholar
4.Arendt, W. and Sourour, A. R., Perturbation of regular operators and the order essential spectrum, Indag. Math. 89 (1986), 109–122.CrossRefGoogle Scholar
5.Barnes, B. A., Inverse closed subalgebras and Fredholm theory, Proc. Roy. Irish Acad. Sect A 83 (1983), 217–224.Google Scholar
6.Barnes, B. A., Murphy, G. J., Smyth, M. R. F. and West, T. T., Riesz and Fredholm theory in Banach algebras, (Pitman, 1982).Google Scholar
7.Bonsall, F. F. and Duncan, J., Complete normed algebras, (Springer-Verlag, 1973).CrossRefGoogle Scholar
8.Dales, H. G. and Woodin, W. H., An introduction to independence for analysts, London Math. Soc. Lecture Notes 115, (Cambridge University Press, 1987).CrossRefGoogle Scholar
9.Groenewald, L., Harte, R. E. and Raubenheimer, H., Perturbation by inessential and Riesz elements, Quaestiones Math., 12 (1989), 439–446.CrossRefGoogle Scholar
10.Harte, R. E., Fredholm theory relative to a Banach algebra homomorphism, Math. Z., 179 (1982), 431–436.CrossRefGoogle Scholar
11.Hocking, J. G. and Young, G. S., Topology (Addison Wesley, 1961).Google Scholar
12.Raubenheimer, H., The o-spectrum of r-asymptotically quasi-finite rank operators, Quaestiones Math., 7 (1984), 299–303.CrossRefGoogle Scholar
13.Raubenheimer, H., r-Asymptotically quasi-finite rank operators and the spectrum of measures, Quaestiones Math., 10 (1986), 97–111.Google Scholar
14.Schaefer, H. H., On the o-spectrum of order bounded operators, Math. Z., 154 (1977), 79–84.CrossRefGoogle Scholar
15.Schep, A. R., Positive diagonal and triangular operators. J. Operator Theory, 3 (1980), 165–178.Google Scholar