Duality results for perturbation classes of semi-Fredholm operators

摘要

The perturbation classes problem for semi-Fredholm operators asks when the equalities SS(X,Y)=PΦ+(X,Y) and SC(X,Y)=PΦ_-(X,Y) are satisfied, where SS and 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, SS(Y*,X*)=PΦ+(Y*,X*) if and only if SC(X,Y)=PΦ-(X,Y), and SC(Y*,X*)=PΦ-(Y*,X*) if and only if SS(X,Y)=PΦ+(X,Y). Moreover we give examples showing that both direct implications fail in general.