  The minimax principle for eigenvalues in gaps of the essential spectrum in
the form presented by Griesemer, Lewis, and Siedentop in [Doc. Math. 4 (1999),
275--283] is adapted to cover certain abstract perturbative settings with
bounded or unbounded perturbations, in particular ones that are off-diagonal
with respect to the spectral gap under consideration. This in part builds upon
and extends the considerations in the author's appendix to [J. Spectr. Theory
10 (2020), 843--885]. Several monotonicity and continuity properties of
eigenvalues in gaps of the essential spectrum are deduced, and the Stokes
operator is revisited as an example.
