Boundedness of adjoint bases of approximate spectral subspaces and of associated block reduced Resolvents

Block reduced resolvents are often employed in iterative schemes for refining crude approximations of the arithmetic mean of a cluster of eigenvalues and of a basis of the corresponding spectral subspace. We prove that if the bases of approximate spectral subspaces are chosen in such a way that they are bounded and each element of the basis is bounded away from the span of the previously chosen elements, then the corresponding adjoint bases are also bounded. We give an integral representation of the associated block reduced resolvent and show that under such a choice of the bases, the approximate block reduced resolvents are bounded as well. This is crucial in obtaining error estimates for the iterates of several refinement schemes. In the framework of a canonical discretization procedure for finite rank operators, appropriate choices of ises are given for various finite rank approximation methods such as Projection, Sloan, Galerkin, Nyström, Fredholm, Degenerate kernel. If the bases are not chosen appropriately, the error estimates may no longer hold and the iteration scheme may not be numerically stable. Examples are given to illustrate these phenomena