From Resolvent Estimates to Semigroup Bounds

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


In Chap.  10 we saw a concrete example of how to get resolvent bounds from semigroup bounds. Naturally, one can go in the opposite direction and in this chapter we discuss some abstract results of that type, including the Hille–Yoshida and Gearhardt–Prüss–Hwang–Greiner theorems. As for the latter, we also give a result of Helffer and the author that provides a more precise bound on the semigroup.


  1. 25.
    N. Burq, M. Zworski, Geometric control in the presence of a black box. J. Am. Math. Soc. 17(2), 443–471 (2004)MathSciNetCrossRefGoogle Scholar
  2. 37.
    E.B. Davies, Semigroup growth bounds. J. Operator Theory 53(2), 225–249 (2005)MathSciNetzbMATHGoogle Scholar
  3. 43.
    K.J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, vol. 194 (Springer, New York, 2000)Google Scholar
  4. 46.
    I. Gallagher, Th. Gallay, F. Nier, Spectral asymptotics for large skew-symmetric perturbations of the harmonic oscillator. Int. Math. Res. Not. IMRN 2009(12), 2147–2199 (2009)MathSciNetzbMATHGoogle Scholar
  5. 57.
    B. Helffer, Spectral Theory and Its Applications. Cambridge Studies in Advanced Mathematics, vol. 139 (Cambridge University Press, Cambridge, 2013)Google Scholar
  6. 62.
    B. Helffer, J. Sjöstrand, From resolvent bounds to semigroup bounds, published as part III of the published version of Ref. [138] (2010).
  7. 66.
    M. Hitrik, Eigenfrequencies and expansions for damped wave equations. Methods Appl. Anal. 10(4), 543–564 (2003)MathSciNetzbMATHGoogle Scholar
  8. 122.
    E. Schenk, Systèmes quantiques ouverts et méthodes semi-classiques, thèse novembre, 2009. t09/266/public/thesis_schenck.pdfGoogle Scholar
  9. 152.
    L.N. Trefethen, M. Embree, Spectra and Pseudospectra. The Behavior of Nonnormal Matrices and Operators (Princeton University Press, Princeton, 2005)Google Scholar
  10. 158.
    G. Weiss, The resolvent growth assumption for semigroups on Hilbert spaces. J. Math. Anal. Appl. 145, 154–171 (1990)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

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

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