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

TL;DR

This paper explores Krylov solvability of inverse problems in Banach spaces, highlighting unique challenges and developing spectral tools to analyze the problem using the resolvent operator.

Contribution

It introduces the first steps in analyzing Krylov solvability in Banach spaces and connects it to spectral theory using resolvent operator techniques.

Findings

Krylov subspace may lack a topological complement in Banach spaces.

Spectral tools are developed to analyze the problem via the resolvent operator.

The approach contrasts with the Hilbert space setting, addressing unique Banach space difficulties.

Abstract

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.