A Note on Spectral Convergence in Varying Hilbert Spaces

TL;DR

This paper establishes conditions under which the spectra of sequences of operators on changing Hilbert spaces converge in the Hausdorff sense, extending previous results on spectral pollution for sectorial operators.

Contribution

It provides new sufficient conditions for spectral convergence of operators on varying Hilbert spaces in the norm-resolvent sense, generalizing earlier work.

Findings

Proves Hausdorff convergence of spectra under norm-resolvent convergence.

Extends spectral convergence results to operators on varying Hilbert spaces.

Addresses spectral pollution issues for sectorial operators.

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.