Krylov Solvability of Unbounded Inverse Linear Problems

Abstract

AbstractThe abstract issue of ‘Krylov solvability’ is extensively discussed for the inverse problem $$Af=g$$ A f = 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 $$g,Ag,A^2g,\dots $$ g , A g , A 2 g , ⋯ , 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.