  We prove sufficient conditions for Hausdorff convergence of the spectra of
sequences of closed operators defined on varying Hilbert spaces and converging
in norm-resolvent sense, i.e. $\|J_\varepsilon(1+A_\varepsilon)^{-1} -
(1+A)^{-1}J_\varepsilon\|\to 0$ as $\varepsilon\to 0$, where $J_\varepsilon$ is
a suitable identification operator between the domains of the operators. This
is an extension of results of [Mugnolo-Nittka-Post(2013)], who proved absence
of spectral pollution for sectorial operators.
