Convergence of Krylov methods for sums of two operators (Q2565274)

The author considers the equation \(x=Tx+f\), where \(T\) is a bounded linear operator in a complex separable Hilbert space \(H\) and \(f\in H\) is given. Assume that \(1-T\) has a bounded inverse and that it is possible to evaluate \(Ty\) at any given vector \(y\in H\). Let \(q\) be any polynomial and put \(p(\lambda):= 1-(1- \lambda)q(\lambda)\). It is known that \(\widehat{x}:= q(T)f\) is a good approximation to \(x=(1-T)^{-1}f\) if and only if \(|p(T)f|\) is small. Krylov subspace methods approximate the solution \(x=(1-T)^{-1}f\) by finite linear combinations of \(T^jf\); these methods are based on projecting, orthogonally or almost, on Krylov subspaces \(K_n(T,f):= \text{span} \{T^jf\}_0^{n-1}\). The convergence speed can be analyzed in terms of \(\min|p_n(T)f|\). The author demonstrates that if \(T=B+K\) where \(B\) is bounded and \(K\) is nuclear, then the obtainable speed for \(T\) is not much slower than the obtainable speeds for \(B\) and \(K\) separately. Bounds are given for \(\inf|Q_n(B+K)|\) and for \(\inf|p_n(B+K)|\), where the infimum is over all polynomials of degree \(n\), such that \(Q_n\) is monic and \(p_n\) is normalized: \(p_n(1)=1\).