  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 $g, Ag, A^2 g, \dots$ , 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.
