Fredholm theory for pairs of closed subspaces of a Banach space

We study the Fredholm theory for pairs of closed subspaces of a Banach space developed by Kato. We define the strictly singular and the strictly cosingular pairs of subspaces, and we show that some of the results of operator theory can be extended to this context. However, there are some notable differences. On the one hand, the perturbation classes problem has a positive answer in this context: the upper and lower semi-Fredholm pairs are stable under strictly singular and strictly cosingular perturbations, respectively, and this stability characterizes the strictly singular and the strictly cosingular pairs. Note that it has been proved recently that the perturbation classes problem for continuous semi-Fredholm operators has a negative answer. On the other hand, unlike in the case of operators, the Fredholm pairs are not stable under perturbation by strictly singular or strictly cosingular pairs. We also show the stability under composition of the compact, the strictly singular and the strictly cosingular pairs of subspaces.