Opuscula Math. 43, no. 1 (2023), 101-108
https://doi.org/10.7494/OpMath.2023.43.1.101
Opuscula Mathematica
A note on Hausdorff convergence of pseudospectra
Marko Lindner
Dennis Schmeckpeper
Abstract. For a bounded linear operator on a Banach space, we study approximation of the spectrum and pseudospectra in the Hausdorff distance. We give sufficient and necessary conditions in terms of pointwise convergence of appropriate spectral quantities.
Keywords: resolvent, spectrum, pseudospectrum, Hausdorff convergence.
Mathematics Subject Classification: 47A10, 47A25.
- S.N. Chandler-Wilde, R. Chonchaiya, M. Lindner, On spectral inclusion sets and computing the spectra and pseudospectra of bounded linear operators, in preparation.
- M.J. Colbrook, Pseudoergodic operators and periodic boundary conditions, Math. Comp. 89 (2020), no. 322, 737-766. https://doi.org/10.1090/mcom/3475
- E.B. Davies, E. Shargorodsky, Level sets of the resolvent norm of a linear operator revisited, Mathematika 62 (2015), no. 1, 243-265. https://doi.org/10.1112/S0025579315000194
- M.M. Day, Some more uniformly convex spaces, Bull. Amer. Math. Soc. 47 (1941), 504-507. https://doi.org/10.1090/S0002-9904-1941-07499-9
- F. Gabel, D. Gallaun, J. Grossmann, M. Lindner, R. Ukena, Spectral approximation of generalized Schrödinger operators via approximation of subwords, in preparation.
- J. Globevnik, Norm-constant analytic functions and equivalent norms, Illinois J. Math. 20 (1976), no. 3, 503-506. https://doi.org/10.1215/ijm/1256049790
- R. Hagen, S. Roch, B. Silbermann, \(C^*\)-Algebras and Numerical Analysis, Monographs and Textbooks in Pure and Applied Mathematics, vol. 236, Marcel Dekker, Inc., New York, Basel, 2001.
- R. Hagger, M. Lindner, M. Seidel, Essential pseudospectra and essential norms of band-dominated operators, J. Math. Anal. Appl. 437 (2016), no. 1, 255-291. https://doi.org/10.1016/j.jmaa.2015.11.060
- M. Lindner, D. Schmeckpeper, How stability indicators determine asymptotics of resolvents, condition numbers and pseudospectra, in preparation.
- M. Lindner, M. Seidel, An affirmative answer to a core issue on limit operators, J. Funct. Anal. 267 (2014), no. 3, 901-917. https://doi.org/10.1016/j.jfa.2014.03.002
- E. Shargorodsky, On the level sets of the resolvent norm of a linear operator, Bull. London Math. Soc. 40 (2008), no. 3, 493-504.
- E. Shargorodsky, On the definition of pseudospectra, Bull. London Math. Soc. 41 (2009), no. 3, 524-534. https://doi.org/10.1112/blms/bdp031
- E. Shargorodsky, S. Shkarin, The level sets of the resolvent norm and convexity properties of Banach spaces, Arch. Math. 93 (2009), no. 1, 59-66. https://doi.org/10.1007/s00013-009-0001-z
- L.N. Trefethen, M. Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, Princeton Univ. Press, Princeton, NJ, 2005.
- Marko Lindner (corresponding author)
- TU Hamburg, Institute of Mathematics, Am Schwarzenberg - Campus 1, 21073 Hamburg, Germany
- Dennis Schmeckpeper
- TU Hamburg, Institute of Mathematics, Am Schwarzenberg - Campus 1, 21073 Hamburg, Germany
- Communicated by P.A. Cojuahri.
- Received: 2022-10-13.
- Revised: 2022-11-01.
- Accepted: 2022-11-05.
- Published online: 2022-12-30.