Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T07:40:41.416Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximating Fredholm operators on a nonseparable Hilbert space

Published online by Cambridge University Press:  18 May 2009

Richard Bouldin
Richard Bouldin
Affiliation:
Department of MathematicsUniversity of GeorgiaAthensGeorgia 30602U.S.A.
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.

This paper obtains a simple formula for the distance from a given operator to the set of invertible operators without requiring the underlying space to be separable. That formula is used to compute the distance to the Fredholm operators with a given index. These results require the further study of the concepts of essential nullity and essential deficiency, which permitted us to characterize the closure of the invertible operators. We also introduce a parameter called the modulus of Fredholmness.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 35 , Issue 2 , May 1993 , pp. 167 - 178
Copyright
Copyright © Glasgow Mathematical Journal Trust 1993

References

REFERENCES

1.Apostol, C., Fialkow, L. A., Herrero, D. A. and Voiculescu, D., Approximation of Hilbert space operators, vol. II (Pitman, Boston, 1984).Google Scholar
2.Bouldin, R. H., The essential minimum modulus, Indiana Univ. Math. J. 30 (1981), 513–517.CrossRefGoogle Scholar
3.Bouldin, R. H., Approximation by operators with fixed nullity, Proc. Amer. Math. Soc. 103 (1988), 141–144.CrossRefGoogle Scholar
4.Bouldin, R. H., The distance to operators with a fixed index, Ada Sci. Math. (Szeged) 54 (1990), 139–143.Google Scholar
5.Bouldin, R. H., Closure of invertible operators on a Hilbert space, Proc. Amer. Math. Soc. 108 (1990), 721–726.CrossRefGoogle Scholar
6.Bouldin, R. H., Approximation by semi-Fredholm operators with fixed nullity, Rocky Mountain J. Math. 20 (1990), 39–50.CrossRefGoogle Scholar
7.Bouldin, R. H., Distance to invertible operators without separability, Proc. Amer. Math. Soc. (to appear).Google Scholar
8.Burlando, L., On continuity of the spectral radius function in Banach algebras, Ann. Mat. Pura Appl. (to appear).Google Scholar
9.Burlando, L., Distance formulas on operators whose kernel has fixed Hilbert dimension, Rendiconti di Matematica 10 (1990), 209–238.Google Scholar
10.Feldman, J. and Kadison, R. V., The closure of the regular operators in a ring of operators, Proc. Amer. Math. Soc. 5 (1954), 909–916.CrossRefGoogle Scholar
11.Harte, R., Regular boundary elements, Proc. Amer. Math. Soc. 99 (1987), 328–330.CrossRefGoogle Scholar
12.Herrero, D. A., Approximation of Hilbert space operators, vol. I (Pitman, Boston, 1982).Google Scholar
13.Izumino, S., Inequalities on operators with index zero, Math. Japonica 23 (1979), 565–572.Google Scholar
14.Izumino, S. and Kato, Y., The closure of invertible operators on a Hilbert space, Ada Sci. Math. (Szeged) 49 (1985), 321–237.Google Scholar
15.Wu, P. Y., Approximation by invertible and noninvertible operators, J. Approximation Theory 56 (1989), 267–276.CrossRefGoogle Scholar