A note on the Krylov solvability of compact normal operators on Hilbert space

TL;DR

This paper investigates the Krylov solvability of inverse problems involving compact normal operators on Hilbert space, providing explicit descriptions of Krylov subspaces and conditions for solvability, thus expanding understanding of Krylov methods in this context.

Contribution

It explicitly characterizes the Krylov subspace for compact normal operators and proves all inverse problems are Krylov solvable when the data is in the operator's range.

Findings

Explicit description of Krylov subspace for compact normal operators

All inverse problems are Krylov solvable if data is in the operator's range

Isomorphism between Krylov subspace and an $L^2$-measure space

Abstract

We analyse the Krylov solvability of inverse linear problems on Hilbert space $\mathcal{H}$ where the underlying operator is compact and normal. Krylov solvability is an important feature of inverse linear problems that has profound implications in theoretical and applied numerical analysis as it is critical to understand the utility of Krylov based methods for solving inverse problems. Our results explicitly describe for the first time the Krylov subspace for such operators given any datum vector $g\in\mathcal{H}$, as well as prove that all inverse linear problems are Krylov solvable provided that $g$ is in the range of such an operator. We therefore expand our knowledge of the class of Krylov solvable operators to include the normal compact operators. We close the study by proving an isomorphism between the closed Krylov subspace for a general bounded normal operator and an $L^2$-measure space based on the scalar spectral measure.