Advertisement

Norm Estimates for a Semigroup Generated by the Sum of Two Operators with an Unbounded Commutator

  • Michael Gil’Email author
Article
  • 24 Downloads

Abstract

Let A be the generator of an analytic semigroup \((e^{At})_{t\ge 0}\) on a Banach space \({\mathcal {X}}\), B be a bounded operator in \({\mathcal {X}}\) and \(K=AB-BA\) be the commutator. Assuming that there is a linear operator S having a bounded left-inverse operator \(S_l^{-1}\), such that \(\int _{0}^{\infty } \Vert Se^{At}\Vert \Vert e^{Bt}\Vert dt<\infty \), and the operator \(KS_l^{-1}\) is bounded and has a sufficiently small norm, we show that \(\int _{0}^{\infty } \Vert e^{(A+B)t}\Vert dt<\infty \), where \((e^{(A+B)t})_{t\ge 0}\) is the semigroup generated by \(A+B\). In addition, estimates for the supremum- and \(L^1\)-norms of the difference \(e^{(A+B)t}-e^{At}e^{Bt}\) are derived.

Keywords

Banach space semigroups perturbations commutator 

Mathematics Subject Classification

47D06 47D60 

Notes

References

  1. 1.
    Adler, M., Bombieri, M., Engel, K.-J.: On perturbations of generators of \(C_0\)-semigroups. Abstract and Applied Analysis, v. 2014, Article ID 213020, 13 pages (2014)Google Scholar
  2. 2.
    Ahiezer, N.I., Glazman, I.M.: Theory of Linear Operators in a Hilbert Space, Pitman Advanced Publishing Program, Boston (1981)Google Scholar
  3. 3.
    Batty, C.J.K.: On a perturbation theorem of Kaiser and Weis. Semigroup Forum 70, 471–474 (2005)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Batty, C.J.K., Krol, S.: Perturbations of generators of \(C_0\)-semigroups and resolvent decay. J. Math. Anal. Appl. 367, 434–443 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Buse, C., Khan, A., Rahmat, G., Saierli, O.: Weak real integral characterizations for exponential stability of semigroups in reflexive spaces. Semigroup Forum 88, 195–204 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Buse, C., Niculescu, C.: A condition of uniform exponential stability for semigroups. Math. Inequal. Appl. 11(3), 529–536 (2008)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Eisner, T.: Stability of Operators and Operator Semigroups, Operator Theory: Advances and Applications, vol. 209. Birkhäuser Verlag, Basel (2010)zbMATHGoogle Scholar
  8. 8.
    Gil’, M.I.: Operator Functions and Localization of Spectra, Lecture Notes In Mathematics, vol. 1830. Springer-Verlag, Berlin (2003)Google Scholar
  9. 9.
    Gil’, M.I.: Semigroups of sums of two operators with small commutators. Semigroup Forum 98(1), 22–30 (2019)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Guo, B., Zwart, H.: On the relation between stability of continuous- and discrete-time evolution equations via the Cayley transform. Integral Equ. Oper. Theory 54, 349–383 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hadd, S.: Unbounded perturbations of \(C_0\)-semigroups on Banach spaces and applications. Semigroup Forum 70(3), 451–465 (2005)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lectures Notes in Mathematics, No 840. Springer-Verlag, New York (1981)CrossRefGoogle Scholar
  13. 13.
    Heymann, R.: Eigenvalues and stability properties of multiplication operators and multiplication semigroups. Math. Nachr. 287(56), 574–584 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Matrai, T.: On perturbations preserving the immediate norm continuity of semigroups. J. Math. Anal. Appl. 341, 961–974 (2008)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Paunonen, L., Zwart, H.: A Lyapunov approach to strong stability of semigroups. Syst. Control Lett. 62, 673–678 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Seifert, C., Wingert, D.: On the perturbation of positive semigroups. Semigroup Forum 91, 495–501 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Weiss, G.: Weak \(L_p\)-stability of linear semigroup on a Hilbert space implies exponential stability. J. Differ. Equ. 76, 269–285 (1988)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsBen Gurion University of the NegevBeer-ShevaIsrael

Personalised recommendations