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

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.