Denjoy–Carleman Differentiable Perturbation of Polynomials and Unbounded Operators

  • Andreas Kriegl
  • Peter W. Michor
  • Armin Rainer


Let \({t\mapsto A(t)}\) for \({t\in T}\) be a C M -mapping with values unbounded operators with compact resolvents and common domain of definition which are self-adjoint or normal. Here C M stands for C ω (real analytic), a quasianalytic or non-quasianalytic Denjoy–Carleman class, C , or a Hölder continuity class C 0,α . The parameter domain T is either \({\mathbb R}\) or \({\mathbb R^n}\) or an infinite dimensional convenient vector space. We prove and review results on C M -dependence on t of the eigenvalues and eigenvectors of A(t).

Mathematics Subject Classification (2010)

26C10 26E10 47A55 


Perturbation theory differentiable choice of eigenvalues and eigenvectors Denjoy–Carleman ultradifferentiable functions 


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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • Andreas Kriegl
    • 1
  • Peter W. Michor
    • 1
  • Armin Rainer
    • 1
  1. 1.Fakultät für MathematikUniversität WienWienAustria

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