Caruso, Noè AngeloMichelangeli, Alessandro2021-02-222021-02-222019https://openscience.sissa.it/handle/1963/35424. The abstract issue of ‘Krylov solvability’ is extensively discussed for the inverse problem Af = g where A is a (possibly unbounded) linear operator on an infinite-dimensional Hilbert space, and g is a datum in the range of A. The question consists of whether the solution f can be approximated in the Hilbert norm by finite linear combinations of g, Ag, A2g,... , and whether solutions of this sort exist and are unique. After revisiting the known picture when A is bounded, we study the general case of a densely defined and closed A. Intrinsic operator-theoretic mechanisms are identified that guarantee or prevent Krylov solvability, with new features arising due to the unboundedness. Such mechanisms are checked in the self-adjoint case, where Krylov solvability is also proved by conjugate-gradient-based techniques.enInverse linear problemsConjugate gradient methodsUnbounded operators on Hilbert spaceSelf-adjoint operatorsKrylov subspacesKrylov solutionArticle