Some results on Krylov solvability in Banach space and connections to spectral theory

This article contains the first steps in a general analysis of the problem of Krylov solvability of the inverse linear problem in a Banach space. In contrast to the well-studied Hilbert space setting, the Banach space setting presents particular difficulties in creating the connection between Krylov solvability and structural properties of the Krylov subspace itself. At the centre of this is the fact that the closed Krylov subspace may not always have a topological complement. We also develop spectral tools in order to attack the problem using the resolvent operator and exploiting its holomorphic properties on the resolvent set.