Hostname: page-component-7b7567c8c-tnzcf Total loading time: 0 Render date: 2026-11-14T15:56:02.950Z Has data issue: false hasContentIssue false
  • English
  • Français

A contribution to the solution of the compact correction problem for operators on a Banach space

Published online by Cambridge University Press:  18 May 2009

Mícheál Ó Searcóid
Mícheál Ó Searcóid
Affiliation:
Roinn na Matamaitice, Coláiste na Hollscoile, Baile Átha Cliath 4, Éire.
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.

We consider the hypothesis that an operator T on a given Banach space can always be perturbed by a compact operator K in such a way that, whenever a complex number A is in the semi-Fredholm region of T + K, then T + K – λ is either bounded below or surjective. The hypothesis has its origin in the work of West [11], who proved it for Riesz operators on Hilbert space. In this paper, we reduce the general Banach space problem to one of considering only operators of a special type, operators which are, in a spectral sense, natural generalizations of the Riesz operators studied by West.

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 31 , Issue 2 , May 1989, pp. 219 - 229
Copyright
Copyright © Glasgow Mathematical Journal Trust 1989

References

1.Allan, G. R., Holomorphic vector-valued functions on a domain of holomorphy, J. Load. Math. Soc. 42 (1967), 509–513.CrossRefGoogle Scholar
2.Apostol, C., The correction by compact perturbation of the singular behavior of operators, Rev. Roumaine Math Pures Appl. 21 (1976), 155–175.Google Scholar
3.Davidson, K. A. and Herrero, D. A., Decomposition of Banach space operators, Indiana Univ. Math. J. 35 (1986), 333–343.CrossRefGoogle Scholar
4.Herrero, D. A., Economical compact perturbations. I: Erasing normal eigenvalues, J. Operator Theory 10 (1983), 289–306.Google Scholar
5.Kato, T., Perturbation theory for linear operators, (Springer-Verlag, New York, 1966).Google Scholar
6.Laffey, T. J. and West, T. T., Fredholm commutators, Proc. R. Irish Acad. 82A (1982), 129–140.Google Scholar
7.Laurie, C. and Radjavi, H., On the West decomposition of Riesz operators, Bull. Lond. Math. Soc. 12 (1980), 130–132.CrossRefGoogle Scholar
8.Searcóid, M. Ó, Nilpotent decomposition of left and right Fredholm elements of a primering, Proc. R. Irish Acad. 87A (1987), 187–209.Google Scholar
9.Searcóid, M. Ó, Uasleathnú ar theoirimí scarúna West agus Stampfli, Proc. R. Irish Acad. 87A (1987), 27–33.Google Scholar
10.Stampfli, J. G., Compact perturbations, normal eigenvalues and a problem of Salinas, J. Lond. Math. Soc. (2) 9 (1974), 165–175.CrossRefGoogle Scholar
11.West, T. T., The decomposition of Riesz operators, Proc. Lond. Math. Soc. (3) 16 (1966), 737–752.CrossRefGoogle Scholar
12.Yood, B., Properties of linear transformations preserved under addition of a completely continuous transformation, Duke Math. J. 18 (1951), 599–612.CrossRefGoogle Scholar