Some problems of the theory of linear equations

There are considered the structural, approximated and spectral properties of Fredholm operators of index $n$ and $(-n)$, acting between Banach spaces $B$ and $D$, where $D$ is isomorphic to the direct sum of $B$ and finite-dimensional space $E$ of dimension $n$. There is demonstrated the role of S. M. Nikol'skii theorem on Fredholm operator in the study of these properties as well as in the issue of solvability equations with boundary inequalities. For boundary value problems which are uniquely solvable, in the case of a separable Hilbert space $B$, based on Schmidt decomposition for a compact operator a scheme of discretization is proposed, and it allows application of an abstract version of Ryaben'kii–Filippov theorem on the relationship of approximation, stability and convergence.