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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

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