Spectrum and Pseudo-Spectrum

  • Johannes Sjöstrand
Part of the Pseudo-Differential Operators book series (PDO, volume 14)


In this book all Hilbert spaces will be assumed to separable for simplicity. In this section we review some basic definitions and properties; we refer to Kato (Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132. Springer, New York, 1966), Reed and Simon (Methods of modern mathematical physics. I. Functional analysis, 2nd edn. Academic, New York, 1980; Methods of modern mathematical physics. II. Fourier analysis, self adjointness. Academic, New York, 1975; Methods of modern mathematical physics. IV. Analysis of operators. Academic, New York, 1978), Riesz and Sz.-Nagy (Leçons d’analyse fonctionnelle, Quatrième édition. Académie des Sciences de Hongrie, Gauthier-Villars, Editeur- Imprimeur-Libraire, Paris; Akadémiai Kiadó, Budapest 1965) for more substantial presentations.


  1. 5.
    A. Aleman, J. Viola, Singular-value decomposition of solution operators to model evolution equations. Int. Math. Res. Not. IMRN 2015(17), 8275–8288 (2014). MathSciNetCrossRefGoogle Scholar
  2. 6.
    A. Aleman, J. Viola, On weak and strong solution operators for evolution equations coming from quadratic operators. J. Spectr. Theory 8(1), 33–121 (2018). MathSciNetCrossRefGoogle Scholar
  3. 22.
    L.S. Boulton, Non-self-adjoint harmonic oscillator, compact semigroups and pseudospectra. J. Operator Theory 47(2), 413–429 (2002)MathSciNetzbMATHGoogle Scholar
  4. 24.
    L. Boutet de Monvel, Hypoelliptic operators with double characteristics and related pseudo-differential operators. Commun. Pure Appl. Math. 27, 585–639 (1974)MathSciNetCrossRefGoogle Scholar
  5. 34.
    E.B. Davies, Pseudospectra, the harmonic oscillator and complex resonances. Proc. Roy. Soc. Lond. A 455, 585–599 (1999)CrossRefGoogle Scholar
  6. 35.
    E.B. Davies, Pseudopectra of differential operators. J. Operator Theory 43, 243–262 (2000)MathSciNetGoogle Scholar
  7. 68.
    M. Hitrik, L. Pravda-Starov, Spectra and semigroup smoothing for non-elliptic quadratic operators. Math. Ann. 344(4), 801–846 (2009)MathSciNetCrossRefGoogle Scholar
  8. 70.
    M. Hitrik, K. Pravda-Starov, Eigenvalues and subelliptic estimates for non-selfadjoint semiclassical operators with double characteristics. Ann. Inst. Fourier (Grenoble) 63(3), 985–1032 (2013)MathSciNetCrossRefGoogle Scholar
  9. 75.
    M. Hitrik, J. Sjöstrand, J. Viola, Resolvent estimates for elliptic quadratic differential operators. Anal. PDE 6(1), 181–196 (2013)MathSciNetCrossRefGoogle Scholar
  10. 77.
    M. Hitrik, K. Pravda-Starov, J. Viola, From semigroups to subelliptic estimates for quadratic operators. Trans. Am. Math. Soc. 370(10), 7391–7415 (2018). MathSciNetCrossRefGoogle Scholar
  11. 84.
    L. Hörmander, An Introduction to Complex Analysis in Several Variables, 3rd edn. North-Holland Mathematical Library, vol. 7 (North-Holland Publishing Co., Amsterdam, 1990)Google Scholar
  12. 85.
    R.A. Horn, C.R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1985)CrossRefGoogle Scholar
  13. 88.
    T. Kato, Perturbation Theory for Linear Operators. Die Grundlehren der mathematischen Wissenschaften, Band, vol. 132 (Springer, New York, 1966)Google Scholar
  14. 109.
    M. Ottobre, G. Pavliotis, K. Pravda-Starov, Exponential return to equilibrium for hypoelliptic quadratic systems. J. Funct. Anal. 262, 4000–4039 (2012)MathSciNetCrossRefGoogle Scholar
  15. 112.
    K. Pravda-Starov, A complete study of the pseudo-spectrum for the rotated harmonic oscillator. J. Lond. Math. Soc. (2) 73(3), 745–761 (2006)MathSciNetCrossRefGoogle Scholar
  16. 114.
    K. Pravda-Starov, Subelliptic estimates for quadratic differential operators. Am. J. Math. 133, 39–89 (2011)MathSciNetCrossRefGoogle Scholar
  17. 116.
    M. Reed, B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis, 2nd edn. (Academic, New York, 1980), Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness (Academic, New York, 1975), Methods of Modern Mathematical Physics. IV. Analysis of Operators (Academic Press, New York, 1978)Google Scholar
  18. 117.
    F. Riesz, B.-Sz. Nagy, Leçons d’Analyse Fonctionnelle, Quatrième édition. Académie des Sciences de Hongrie, Gauthier-Villars (Editeur-Imprimeur-Libraire/Akadémiai Kiadó, Paris/Budapest, 1965)Google Scholar
  19. 131.
    J. Sjöstrand, Parametrices for pseudodifferential operators with multiple characteristics. Ark. Mat. 12(1), 85–130 (1974)MathSciNetCrossRefGoogle Scholar
  20. 151.
    L.N. Trefethen, Pseudospectra of linear operators. SIAM Rev. 39(3), 383–406 (1997)MathSciNetCrossRefGoogle Scholar
  21. 152.
    L.N. Trefethen, M. Embree, Spectra and Pseudospectra. The Behavior of Nonnormal Matrices and Operators (Princeton University Press, Princeton, 2005)Google Scholar
  22. 153.
    J. Viola, Spectral projections and resolvent bounds for partially elliptic quadratic differential operators. J. Pseudo-Differ. Oper. Appl. 4, 145–221 (2013)MathSciNetCrossRefGoogle Scholar
  23. 154.
    J. Viola, The norm of the non-self-adjoint harmonic oscillator semigroup. Integr. Equ. Oper. Theory 85(4), 513–538 (2016)MathSciNetCrossRefGoogle Scholar
  24. 162.
    M. Zworski, Numerical linear algebra and solvability of partial differential equations. Commun. Math. Phys. 229(2), 293–307 (2002)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Johannes Sjöstrand
    • 1
  1. 1.Université de Bourgogne Franche-ComtéDijonFrance

Personalised recommendations