Optimization of adaptive direct methods for the solution of operator equations in Hilbert space (Q1812898)

Let \(X\) and \(F_ n\) \((\dim F_ n= n)\) be Hilbert spaces and \(F_ n\subset X\) be a subspace of \(X\); let \(H\) and \(H_ n\) be bounded linear operators, \(X\to X\) and \(X\to F_ n\), respectively, and let \(P_ n\) be a projector of \(X\) onto \(F_ n\). The proximity of solutions of the equations \(z= Hz+ f\) and \(z= H_ n z+f\), where \(H_ n= P_ n H+ HP_ n- P_ n HP_ n\), \(z\in X\), is investigated. Under some conditions the extremal properties of \(H_ n\) are proved.