Krylov Solvability of Unbounded Inverse Linear Problems
Noè Angelo Caruso, Alessandro Michelangeli · Integral Equations and Operator Theory · 2020
Abstract The abstract issue of ‘Krylov solvability’ is extensively discussed for the inverse problem $$Af=g$$ Af=g whereAis a (possibly unbounded) linear operator on an infinite-dimensional Hilbert space, andgis a datum in the range ofA. The question consists of whether the solutionfcan be approximated in the Hilbert norm by finite linear combinations of $$g,Ag,A^2g,\dots $$ g,Ag,A2g,⋯ , and whether solutions of this sort exist and are unique. After revisiting the known picture whenAis bounded, we study the general case of a densely defined and closedA. 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.