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Generalized Semi-Fredholm transformations

Part of: General theory of linear operators

Published online by Cambridge University Press:  09 April 2009

D. G. Tacon
D. G. Tacon
Affiliation:
University of New South WalesBox 1, Post Office Kensington, N.S.W., 2033, Australia
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Abstract

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The class ϕ+(X, Y) of semi-Fredholm transformations consists of those transformations T: X → Y for which α(T) = dim ker T < ∞ and for which T(X) is closed. It forms an open subset of B (X, Y) closed under perturbation by compact transformations and is a particularly important class of transformation since T is Fredholm if and only if T ∈ ϕ+ (X, Y) and T′ ∈ ϕ+ (Y′. X′). The realization that elements of ϕ+ (X, Y) have very simple nonstandard characterizations lead the author to consider the possibility of finding an analogous open class of transformation which is closed under perturbation by weakly compact transformations. Consequently this paper investigates two related classes which contain ϕ+ (X, Y). The first such class coincides with the class of Tauberian transformations whilst the second consists of those transformations which have Tauberian extensions on the nonstandard hulls. The Tauberian transformations are closed under perturbation by weakly compact transformations but in general are not open. The “super” Tauberian transformations are closed under perturbation by super weakly compact transformations and in fact form an open subset of B(X, Y).

MSC classification

Secondary: 47A53: (Semi-) Fredholm operators; index theories
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 34 , Issue 1 , February 1983 , pp. 60 - 70
Copyright
Copyright © Australian Mathematical Society 1983

References

1.Caradus, S. R.. Pfaffenberger, W. E. and Yood, Bertram, Calkin algebras and algebras of operators on Banach spaces (Lecture Notes in Pure and Applied Mathematics, Dekker, New York, 1974).Google Scholar
2.Chadwick, J. J. M. and Wickstead, A. W., ‘A quotient of ultrapowers of Banach spaces and semi-Fredholm operators’, Bull. London Math. Soc. 9 (1977), 321–325.CrossRefGoogle Scholar
3.Henson, C. W. and Moore, L. C. Jr, ‘The nonstandard theory of topological vector spaces’. Trans. Amer. Math. Soc. 172 (1972), 405–435,CrossRefGoogle Scholar
Erratum, Trans. Amer. Math. Soc. 184 (1973), 509.Google Scholar
4.James, R. C.. ‘Weakly compact sets’, Trans. Amer. Math. Soc. 113 (1964), 129–140.CrossRefGoogle Scholar
5.James, R. C.. ‘Some self-dual properties of normed linear spaces’. Sympos. on Infinite Dimensional Topology, pp. 159–175 (Ann. of Math. Studies 69, Princeton Univ. Press, Princeton. 1972).CrossRefGoogle Scholar
6.Kalton, N. and Wilansky, A., ‘Tauberian operators on Banach spaces’, Proc. A mer. Math. Soc. 57 (1976), 251–255.CrossRefGoogle Scholar
7.Lebow, A. and Schechter, M., ‘Semigroups of operators and measures of noncompactness’, J. Functional Analysis 7 (1971), 1–26.CrossRefGoogle Scholar
8.Luxemburg, W. A. J., ‘A general theory of monads’, Application of Model Theory to Algebra, Analysis and Probability, Internat. Sympos., Pasadena, Calif., 1967, pp. 18–86, (Holt, Rinehart and Winston, New York, 1969).Google Scholar
9.Luxemburg, W. A. J., ‘On some concurrent binary relations occurring in analysis’, Contributions to nonstandard analysis, pp. 85–100 (Studies in Logic and the Foundations of Math., North-Holland, Amsterdam, London, 1972).CrossRefGoogle Scholar
10.Robinson, A., Non-standard analysis (Studies in Logic and the Foundations of Math., North-Holland, Amsterdam, 1966).Google Scholar
11.Tacon, D. G., ‘Nonstandard extensions of transformations between Banach spaces’, Trans. Amer. Math. Soc. 260 (1980), 147–158.CrossRefGoogle Scholar
12.Yang, K. W., ‘The generalized Fredholm operators’, Trans. Amer. Math. Soc. 216 (1976), 313–326.CrossRefGoogle Scholar
13.Yang, K. W., ‘Operators invertible modulo the weakly compact operators’, Pacific J. Math. 71 (1977), 559–564.CrossRefGoogle Scholar