2020/01/22 by Caruso, Noe, Michelangeli, Alessandro · 1 citation
#FOS: Mathematics #Functional Analysis (math.FA) #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2001.08127
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, A2 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.