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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anselone, P.M., Palmer, T.W.: Spectral analysis of collectively compact, strongly convergent operator sequences. Pac. J. Math. 25, 423–431 (1968)
Bailey, P.B., Everitt, W.N., Weidmann, J., Zettl, A.: Regular approximations of singular Sturm–Liouville problems. Results Math. 23(1–2), 3–22 (1993)
Bögli, S.: Spectral approximation for linear operators and applications. Ph.D. thesis (2014)
Bögli, S., Siegl, P.: Remarks on the convergence of pseudospectra. Integral Equ. Oper. Theory 80(3), 303–321 (2014)
Bögli, S., Siegl, P., Tretter, C.: Approximations of spectra of Schrödinger operators with complex potentials on \({\mathbb{R}}^d\). Commun. Partial Differ. Equ. (2017). doi:10.1080/03605302.2017.1330342
Brown, B.M., Langer, M., Marletta, M., Tretter, C., Wagenhofer, M.: Eigenvalue enclosures and exclosures for non-self-adjoint problems in hydrodynamics. LMS J. Comput. Math. 13, 65–81 (2010)
Brown, B.M., Marletta, M.: Spectral inclusion and spectral exactness for singular non-self-adjoint Sturm–Liouville problems. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457(2005), 117–139 (2001)
Brown, B.M., Marletta, M.: Spectral inclusion and spectral exactness for singular non-self-adjoint Hamiltonian systems. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 459(2036), 1987–2009 (2003)
Brown, B.M., Marletta, M.: Spectral inclusion and spectral exactness for PDEs on exterior domains. IMA J. Numer. Anal. 24, 21–43 (2004)
Chatelin, F.: Spectral Approximation of Linear Operators. Computer Science and Applied Mathematics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York (1983). With a foreword by P. Henrici, with solutions to exercises by Mario Ahués
Davies, E.B.: Spectral enclosures and complex resonances for general self-adjoint operators. LMS J. Comput. Math. 1, 42–74 (1998)
Davies, E.B., Plum, M.: Spectral pollution. IMA J. Numer. Anal. 24(3), 417–438 (2004)
Edmunds, D.E., Evans, W.D.: Spectral Theory and Differential Operators. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1987)
Grigorieff, R.D.: Diskret kompakte Einbettungen in Sobolewschen Räumen. Math. Ann. 197, 71–85 (1972)
Hess, P., Kato, T.: Perturbation of closed operators and their adjoints. Comment. Math. Helv. 45, 524–529 (1970)
Hinchcliffe, J., Strauss, M.: Spectral enclosure and superconvergence for eigenvalues in gaps. Integral Equ. Oper. Theory 84(1), 1–32 (2016)
Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1995). Reprint of the 1980 edition
Leoni, G.: A First Course in Sobolev Spaces, vol. 105 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2009)
Levitin, M., Shargorodsky, E.: Spectral pollution and second-order relative spectra for self-adjoint operators. IMA J. Numer. Anal. 24(3), 393–416 (2004)
Mertins, U., Zimmermann, S.: Variational bounds to eigenvalues of self-adjoint eigenvalue problems with arbitrary spectrum. Z. Anal. Anwend. 14(2), 327–345 (1995)
Osborn, J.E.: Spectral approximation for compact operators. Math. Comput. Simul. 29, 712–725 (1975)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics. I. Functional Analysis. Academic Press, New York (1972)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press, New York (1978)
Shargorodsky, E.: Geometry of higher order relative spectra and projection methods. J. Oper. Theory 44(1), 43–62 (2000)
Stummel, F.: Diskrete Konvergenz linearer Operatoren. I. Math. Ann. 190, 45–92 (1970)
Stummel, F.: Diskrete Konvergenz linearer Operatoren. II. Math. Z. 120, 231–264 (1971)
Tretter, C.: Spectral Theory of Block Operator Matrices and Applications. Imperial College Press, London (2008)
Weidmann, J.: Lineare Operatoren in Hilberträumen. Teil I. Mathematische Leitfäden. [Mathematical Textbooks]. B. G. Teubner, Stuttgart (2000). Grundlagen. [Foundations]
Weidmann, J.: Lineare Operatoren in Hilberträumen. Teil II. Mathematische Leitfäden. [Mathematical Textbooks]. B. G. Teubner, Stuttgart (2003). Anwendungen. [Applications]
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00020-017-2389-3