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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)



Perturbation theory for normal operators

Author: Armin Rainer
Journal: Trans. Amer. Math. Soc. 365 (2013), 5545-5577
MSC (2010): Primary 26C10, 26E10, 32B20, 47A55
Published electronically: February 19, 2013
MathSciNet review: 3074382
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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.

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Additional Information

Armin Rainer
Affiliation: Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, A-1090 Wien, Austria

Keywords: Perturbation theory, differentiable and Lipschitz eigenvalues and eigenvectors, normal unbounded operators, resolution of singularities, Denjoy–Carleman classes
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
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.