Structural properties of Krylov subspaces, Krylov solvability, and applications to unbounded self-adjoint operators

TL;DR

This paper investigates the structural properties of Krylov subspaces for self-adjoint operators, exploring Krylov solvability in unbounded settings and linking approximation properties to the Hamburger moment problem.

Contribution

It introduces new insights into Krylov subspace structures for unbounded self-adjoint operators and connects these properties with classical moment problem theory.

Findings

Krylov subspaces exhibit specific structural properties for self-adjoint operators.

Krylov solvability relates to the approximation capabilities of these subspaces.

Connections are established between Krylov approximation and the Hamburger moment problem.

Abstract

This paper presents a study of the inherent structural properties of Krylov subspaces, in particular for the self-adjoint class of operators, and how they relate with the important phenomenon of `Krylov solvability' of linear inverse problems. Owing to the complexity of the problem in the unbounded setting, recently developed perturbative techniques are used that exploit the use of the weak topology on $\mathcal{H}$. We also make a strong connection between the approximation properties of the Krylov subspace and the famous Hamburger problem of moments, in particular the determinacy condition.