Duality results for perturbation classes of semi-Fredholm operators
The summary perfectly describes this paper: ``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 \(P\Phi _+\) and \(P\Phi _-\) denote the perturbation classes for upper semi-Fredholm and lower semi-Fredholm operators. We show that, when \(Y\) is a reflexive Banach space, \(\mathcal{SS}(X^*,Y^*)=P\Phi_+(X^*,Y^*)\) if and only if \(\mathcal{SC}(X,Y)=P\Phi _-(X,Y)\), and \(\mathcal{SC}(X^*,Y^*)=P\Phi _-(X^*,Y^*)\) if and only if \(\mathcal{SS}(X,Y)=P\Phi_+(X,Y)\). Moreover, we give examples showing that both direct implications fail in general.''
scientific article; zbMATH DE number 5988048
scientific article