Perturbation theory for normal operators
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Abstract:
Let $E \ni x\mapsto A(x)$ be a $\mathscr {C}$-mapping with its values being unbounded normal operators with common domain of definition and compact resolvent. Here $\mathscr {C}$ stands for $C^\infty$, $C^\omega$ (real analytic), $C^{[M]}$ (Denjoy–Carleman of Beurling or Roumieu type), $C^{0,1}$ (locally Lipschitz), or $C^{k,\alpha }$. The parameter domain $E$ is either $\mathbb R$ or $\mathbb R^n$ or an infinite dimensional convenient vector space. We completely describe the $\mathscr {C}$-dependence on $x$ of the eigenvalues and the eigenvectors of $A(x)$. Thereby we extend previously known results for self-adjoint operators to normal operators, partly improve them, and show that they are best possible. For normal matrices $A(x)$ we obtain partly stronger results.References
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Additional Information
- Armin Rainer
- Affiliation: Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, A-1090 Wien, Austria
- MR Author ID: 752266
- ORCID: 0000-0003-3825-3313
- Email: armin.rainer@univie.ac.at
- Received by editor(s): December 21, 2011
- Received by editor(s) in revised form: April 2, 2012
- Published electronically: February 19, 2013
- Additional Notes: This work was supported by the Austrian Science Fund (FWF), Grant P 22218-N13
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 5545-5577
- MSC (2010): Primary 26C10, 26E10, 32B20, 47A55
- DOI: https://doi.org/10.1090/S0002-9947-2013-05854-0
- MathSciNet review: 3074382