  We introduce compactness classes of Hilbert space operators by grouping
together all operators for which the associated singular values decay at a
certain speed and establish upper bounds for the norm of the resolvent of
operators belonging to a particular compactness class. As a consequence we
obtain explicitly computable upper bounds for the Hausdorff distance of the
spectra of two operators belonging to the same compactness class in terms of
the distance of the two operators in operator norm.
