  We propose an a posteriori error estimator for high-order $p$- or $hp$-finite
element discretizations of selfadjoint linear elliptic eigenvalue problems that
is appropriate for estimating the error in the approximation of an eigenvalue
cluster and the corresponding invariant subspace. The estimator is based on the
computation of approximate error functions in a space that complements the one
in which the approximate eigenvectors were computed. These error functions are
used to construct estimates of collective measures of error, such as the
Hausdorff distance between the true and approximate clusters of eigenvalues,
and the subspace gap between the corresponding true and approximate invariant
subspaces. Numerical experiments demonstrate the practical effectivity of the
approach.
