Local convergence of spectra and pseudospectra

Bögli, Sabine

doi:10.4171/JST/222

Journal of Spectral Theory

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Volume 8, Issue 3, 2018, pp. 1051–1098

DOI: 10.4171/JST/222

Published online: 2018-07-12

Local convergence of spectra and pseudospectra

Sabine Bögli^[1]

(1) Ludwig-Maximilians-Universität München, Germany

We prove local convergence results for the spectra and pseudospectra of sequences of linear operators acting in different Hilbert spaces and converging in generalised strong resolvent sense to an operator with possibly non-empty essential spectrum. We establish local spectral exactness outside the limiting essential spectrum, local $\varepsilon$-pseudospectral exactness outside the limiting essential $\varepsilon$-near spectrum, and discuss properties of these two notions including perturbation results.

Keywords: Eigenvalue approximation, spectral exactness, spectral inclusion, spectral pollution, resolvent convergence, pseudospectra

Bögli Sabine: Local convergence of spectra and pseudospectra. J. Spectr. Theory 8 (2018), 1051-1098. doi: 10.4171/JST/222