Fredholm theory for pairs of closed subspaces of a Banach space
\textit{T. Kato} in chapter IV of [``Perturbation theory for linear operators'' (Springer-Verlag Berlin) (1995; Zbl 0836.47009)] developed a perturbation theory for upper and lower semi-Fredholm pairs of subspaces of a Banach space. The present author introduces the notions of compact, strictly singular and strictly cosingular pairs of subspaces of a Banach space. He proves that a pair of subspaces \((L,M)\) is strictly singular if and only if \((L,N)\) is upper semi-Fredholm for every closed subspace \(N\) such that \((M,N)\) is upper semi-Fredhom. Moreover, he demonstrates that the upper semi-Fredholmness of \((L,M)\) and strict singularity of \((M,N)\) do not imply the upper semi-Fredholmness of \((L,N)\). Counterparts of these results for lower semi-Fredholmness and strict cosingularity are also obtained. Finally, the author proves that the classes of compact, strictly singular, and strictly cosingular pairs are transitive.
scientific article; zbMATH DE number 2159772
scientific article