  When a solution to an abstract inverse linear problem on Hilbert space is
approximable by finite linear combinations of vectors from the cyclic subspace
associated with the datum and with the linear operator of the problem, the
solution is said to be a Krylov solution, i.e., it belongs to the Krylov
subspace of the problem. Krylov solvability of the inverse problem allows for
solution approximations that, in applications, correspond to the very efficient
and popular Krylov subspace methods. We study here the possible behaviours of
persistence, gain, or loss of Krylov solvability under suitable small
perturbations of the inverse problem -- the underlying motivations being the
stability or instability of Krylov methods under small noise or uncertainties,
as well as the possibility to decide a priori whether an inverse problem is
Krylov solvable by investigating a potentially easier, perturbed problem. We
present a whole scenario of occurrences in the first part of the work. In the
second, we exploit the weak gap metric induced, in the sense of Hausdorff
distance, by the Hilbert weak topology, in order to conveniently monitor the
distance between perturbed and unperturbed Krylov subspaces.
