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Generlized 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 WalesP.O. Box 1 Kensington, N.S.W. 2033, Australia
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Abstract

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In an earlier paper we showed that the set ψ+ (X, Y) of super Tauberian transformations between two Banach spaces X and Y forms an open subset of B(X, Y) which is closed under perturbation by super weakly compact transformations. In this note we characterize a class dual to ψ+ (X, Y) which we denote by ψ-(X, Y). We show that T∈ψ+(X, Y) if and only if T′ ∈ ψ-(Y′, X′) and that T′∈ψ+(Y′, X′) if and only if T ∈ ψ-(X, Y) and provide standard and nonstandard characterizations of elements of ψ-(X, Y). These two classes thus play in some ways analogous roles to the sets of semi-Fredholm transforms ϕ+ (X, Y) and ϕ-(X, Y).

Moreover en forms an open subset of B(X, Y) closed under the taking of adjoints, under the taking of nonstandard hull extensions, and under perturbation by super weakly compact transformations.

MSC classification

Secondary: 47A53: (Semi-) Fredholm operators; index theories
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 37 , Issue 1 , August 1984 , pp. 89 - 97
Copyright
Copyright © Australian Mathematical Society 1984

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