Hostname: page-component-7c8c6479df-r7xzm Total loading time: 0 Render date: 2024-03-28T15:59:02.598Z Has data issue: false hasContentIssue false

A spectral problem in ordered Banach algebras

Published online by Cambridge University Press:  17 April 2009

S. Mouton
Affiliation:
Department of Mathematics, University of Stellenbosch, Private Bag X1, Matieland 7602, South Africa, e-mail: smo@sun.ac.za
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.

We recall the definition and properties of an algebra cone C of a complex unital Banach algebra A. It can be shown that C induces on A an ordering which is compatible with the algebraic structure of A, and A is then called an ordered Banach algebra. The Banach algebra ℒ(E) of all bounded linear operators on a complex Banach lattice E is an example of an ordered Banach algebra, and an interesting aspect of research in ordered Banach algebras is that of investigating in an ordered Banach algebra-context certain problems that originated in ℒ(E). In this paper we investigate the problems of providing conditions under which (1) a positive element a with spectrum consisting of 1 only will necessarily be greater than or equal to 1, and (2) f (a) will be positive if a is positive, where f (a) is the element defined by the holomorphic functional calculus.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2003

References

REFERENCES

[1]Aupetit, B., A primer on spectral theory (Springer-Verlag, New York, Heidelberg, Berlin, 1991).CrossRefGoogle Scholar
[2]Bonsall, F.F. and Duncan, J., Complete normed algebras (Springer-Verlag, New York, Heidelberg, Berlin, 1973).CrossRefGoogle Scholar
[3]Grobler, J.J. and Huijsmans, C.B., ‘Doubly Abel bounded operators with single spectrum’, Quaestiones Math. 18 (1995), 397406.CrossRefGoogle Scholar
[4]Mbekhta, M. and Zemánek, J., ‘Sur le théorème ergodique uniforme et le spectre’, C.R. Acad. Sci. Paris Sér. I Math. 317 (1993), 11551158.Google Scholar
[5]Mouton, H. du T. and Mouton, S., ‘Domination properties in ordered Banach algebras’, Studia Math. 149 (2002), 6373.CrossRefGoogle Scholar
[6]Mouton, S., ‘Convergence properties of positive elements in Banach algebras’, Proc. Roy. Irish Acad. Sect. A (to appear).Google Scholar
[7]Mouton, S. (née Rode) and Raubenheimer, H., ‘More spectral theory in ordered Banach algebras’, Positivity 1 (1997), 305317.CrossRefGoogle Scholar
[8]Raubenheimer, H. and Rode, S., ‘Cones in Banach algebras’, Indag. Math. (N.S.) 7 (1996), 489502.CrossRefGoogle Scholar
[9]Schaefer, H.H., ‘Some spectral properties of positive linear operators’, Pacific J. Math. 10 (1960), 10091019.CrossRefGoogle Scholar
[10]Schaefer, H.H., Wolff, M. and Arendt, W., ‘On lattice isomorphisms with positive real spectrum and groups of positive operators’, Math. Z. 164 (1978), 115123.CrossRefGoogle Scholar
[11]Zhang, X.-D., ‘Some aspects of the spectral theory of positive operators’, Acta Appl. Math. 27 (1992), 135142.CrossRefGoogle Scholar
[12]Zhang, X.-D., ‘On spectral properties of positive operators’, Indag. Math. (N.S.) 4 (1993), 111127.CrossRefGoogle Scholar