Advertisement

Semigroup Forum

, Volume 92, Issue 3, pp 587–597 | Cite as

A note on the periodic decomposition problem for semigroups

  • Bálint FarkasEmail author
RESEARCH ARTICLE
  • 136 Downloads

Abstract

Given \(T_1,\dots , T_n\) commuting power-bounded operators on a Banach space we study under which conditions the equality \(\ker (T_1-\mathrm {I})\cdots (T_n-\mathrm {I})=\ker (T_1-\mathrm {I})+\cdots +\ker (T_n-\mathrm {I})\) holds true. This problem, known as the periodic decomposition problem, goes back to I. Z. Ruzsa. In this short note we consider the case when \(T_j=T(t_j), t_j>0, j=1,\dots , n\) for some one-parameter semigroup \((T(t))_{t\ge 0}\). We also look at a generalization of the periodic decomposition problem when instead of the cyclic semigroups \(\{T_j^n:n \in \mathbb {N}\}\) more general semigroups of bounded linear operators are considered.

Keywords

Periodic decomposition problem \(C_0\)-(semi)group  Norm-continuous one-parameter semigroup Amenable semigroup 

Notes

Acknowledgments

The author was supported by the Hungarian Research Fund (OTKA-100461).

References

  1. 1.
    Basit, R.B.: Generalization of two theorems of M.I. Kadets concerning the indefinite integral of abstract almost periodic functions. Mat. Zametki 9(3), 311–321 (1971)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Eisner, T., Farkas, B., Haase, M., Nagel, R.: Operator Theoretic Aspects of Ergodic Theory, Graduate Texts in Mathematics, vol. 272, Springer-Verlag, New York (2015)Google Scholar
  3. 3.
    Engel, K.-J., Nagel, R.: One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York (2000)Google Scholar
  4. 4.
    Farkas, B.: A Bohl-Bohr-Kadets type theorem characterizing Banach spaces not containing \(c_0\). arxiv.org/abs/1301.6250 (2013). (submitted)
  5. 5.
    Farkas, B., Révész, Sz.: Decomposition as the sum of invariant functions with respect to commuting transformations. Aequ. Math 73(3), 233–248 (2007)Google Scholar
  6. 6.
    Farkas, B., Harangi, V., Keleti, T., Révész, Sz.: Invariant decomposition of functions with respect to commuting invertible transformations. Proc. Am. Math. Soc. 136(4), 1325–1336 (2008)Google Scholar
  7. 7.
    Gajda, Z.: Note on decomposition of bounded functions into the sum of periodic terms. Acta Math. Hung. 59(1–2), 103–106 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kadets, V.M., Shumyatskiy, B.M.: Averaging technique in the periodic decomposition problem. Mat. Fiz. Anal. Geom. 7(2), 184–195 (2000)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Kadets, V.M., Shumyatskiy, B.M.: Additions to the periodic decomposition theorem. Acta Math. Hung. 90(4), 293–305 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Laczkovich, M., Révész, Sz.: Periodic decompositions of continuous functions. Acta Math. Hung. 54(3–4), 329–341 (1989)Google Scholar
  11. 11.
    Laczkovich, M., Révész, Sz.: Decompositions into the sum of periodic functions belonging to a given Banach space. Acta Math. Hung. 55(3–4), 353–363 (1990)Google Scholar
  12. 12.
    Martinez, J., Mazon, J.M.: \(C_0\)-semigroups norm continuous at infinity. Semigroup Forum 52(2), 213–224 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Nagel, R.J.: Mittelergodische Halbgruppen linearer operatoren. Ann. Inst. Fourier (Grenoble) 23(4), 75–87 (1973)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Faculty of Mathematics and Natural SciencesUniversity of WuppertalWuppertalGermany

Personalised recommendations