On Krylov Methods in Infinite-dimensional Hilbert Space

This thesis contains the development of key features for the solution to inverse linear problems Af = g on infinite-dimensional Hilbert space H using projection methods. Particular attention is paid to Krylov subspace methods. Intrinsic, key operator-theoretic constructs that guarantee the ‘Krylov solvability’ of the problem Af = g are developed and investigated for this class of projection methods. This theory is supported by numerous examples, counterexamples, and some numerical tests. Results for both bounded and unbounded operators on general Hilbert spaces are considered, with special attention paid to the Krylov method of conjugate-gradients in the unbounded setting.