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- Title
Duality results for perturbation classes of semi-Fredholm operators.
- Authors
González, Manuel
- Abstract
The perturbation classes problem for semi-Fredholm operators asks when the equalities $${\mathcal{SS}(X,Y)=P\Phi_+(X,Y)}$$ and $${\mathcal{SC}(X,Y)=P\Phi_-(X,Y)}$$ are satisfied, where $${\mathcal{SS}}$$ and $${\mathcal{SC}}$$ denote the strictly singular and the strictly cosingular operators, and PΦ and PΦ denote the perturbation classes for upper semi-Fredholm and lower semi-Fredholm operators. We show that, when Y is a reflexive Banach space, $${\mathcal{SS}(Y^*,X^*)=P\Phi_+(Y^*,X^*)}$$ if and only if $${\mathcal{SC}(X,Y)=P\Phi_-(X,Y),}$$ and $${\mathcal{SC}(Y^*,X^*)=P\Phi_-(Y^*,X^*)}$$ if and only if $${\mathcal{SS}(X,Y)=P\Phi_+(X,Y)}$$. Moreover we give examples showing that both direct implications fail in general.
- Subjects
PERTURBATION theory; FREDHOLM operators; FREDHOLM equations; SET theory; BANACH spaces
- Publication
Archiv der Mathematik, 2011, Vol 97, Issue 4, p345
- ISSN
0003-889X
- Publication type
Article
- DOI
10.1007/s00013-011-0313-7