Cluster robust error estimates for the Rayleigh–Ritz approximation I: Estimates for invariant subspaces

This is the first part of a paper that deals with error estimates for the Rayleigh–Ritz approximations to the spectrum and invariant subspaces of a bounded Hermitian operator in a Hilbert or Euclidean space. This part addresses estimates for the angles between the invariant subspaces and their approximations via the corresponding best approximation errors and residuals and, for invariant subspaces corresponding to parts of the discrete spectrum, via eigenvalue errors. The paper’s major concern is to ensure that the estimates in question are accurate and ‘cluster robust’, i.e. are not adversely affected by the presence of clustered, i.e. closely situated eigenvalues in the spectrum. Available estimates of such kind are reviewed and new estimates are derived. The paper’s main new results introduce estimates for invariant subspaces in which the operator may have clustered eigenvalues whereby not only the distances between eigenvalues in the cluster are not present but also the distances between the cluster and the rest of the spectrum appear in asymptotically insignificant terms only.