  We set up, at the abstract Hilbert space setting, the general question on
when an inverse linear problem induced by an operator of Friedrichs type admits
solutions belonging to (the closure of) the Krylov subspace associated to such
operator. Such Krylov solvability of abstract Friedrichs systems allows to
predict when, for concrete differential inverse problems, truncation algorithms
can or cannot reproduce the exact solutions in terms of approximants from the
Krylov subspace.
