A Note on Spectral Convergence in Varying Hilbert Spaces

Abstract: 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.