Convergence of Sequences of Linear Operators and Their Spectra

Abstract

We establish spectral convergence results of approximations of unbounded non-selfadjoint linear operators with compact resolvents by operators that converge in generalized strong resolvent sense. The aim is to establish general assumptions that ensure spectral exactness, i.e. that every true eigenvalue is approximated and no spurious eigenvalues occur. A main ingredient is the discrete compactness of the sequence of resolvents of the approximating operators. We establish sufficient conditions and perturbation results for strong convergence and for discrete compactness of the resolvents.

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Acknowledgements

The author would like to thank her doctoral advisor Christiane Tretter for the guidance. The work was supported by the Swiss National Science Foundation (SNF), Grant No. 200020_146477 and Early Postdoc.Mobility Project P2BEP2_159007.

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Correspondence to Sabine Bögli.

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Bögli, S. Convergence of Sequences of Linear Operators and Their Spectra. Integr. Equ. Oper. Theory 88, 559–599 (2017). https://doi.org/10.1007/s00020-017-2389-3

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Mathematics Subject Classification

  • 47A10
  • 47A55
  • 47A58
  • 47B07

Keywords

  • Eigenvalue approximation
  • Spectral exactness
  • Generalized resolvent convergence
  • Discrete compactness