Some spectral properties of linear operators on exotic banach spaces
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Date
2012
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LOBACHEVSKII JOURNAL OFMATHEMATICS
Abstract
In this work, we present some results concerning the operators defined on various
classes of exotic Banach spaces, containing in particular those studied respectively by V. Ferenczi [7,
8] and T. Gowers with B.Maurey [14, 15].We show that, on hereditarily indecomposable or quotient
hereditarily indecomposable Banach space X, the set of bounded Fredholm operators is dense in
L(X), this gives that the boundary of bounded Fredholm operators is nothing else but the ideal of
strictly singular operators ifX is hereditarily indecomposable Banach space (resp. the ideal of strictly
cosingular operators ifX is quotient hereditarily indecomposable Banach space).On the other hand,
a comparison between sufficiently rich and exotic Banach spaces is given via some properties of the
two maps spectra andWolf essential spectra.
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Keywords
Fredholm perturbation, semi-Fredholm operator, Fredholm operator, hereditarily indecomposable Banach space, essential spectrum