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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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)
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)
Engel, K.-J., Nagel, R.: One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York (2000)
Farkas, B.: A Bohl-Bohr-Kadets type theorem characterizing Banach spaces not containing \(c_0\). arxiv.org/abs/1301.6250 (2013). (submitted)
Farkas, B., Révész, Sz.: Decomposition as the sum of invariant functions with respect to commuting transformations. Aequ. Math 73(3), 233–248 (2007)
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)
Gajda, Z.: Note on decomposition of bounded functions into the sum of periodic terms. Acta Math. Hung. 59(1–2), 103–106 (1992)
Kadets, V.M., Shumyatskiy, B.M.: Averaging technique in the periodic decomposition problem. Mat. Fiz. Anal. Geom. 7(2), 184–195 (2000)
Kadets, V.M., Shumyatskiy, B.M.: Additions to the periodic decomposition theorem. Acta Math. Hung. 90(4), 293–305 (2001)
Laczkovich, M., Révész, Sz.: Periodic decompositions of continuous functions. Acta Math. Hung. 54(3–4), 329–341 (1989)
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)
Martinez, J., Mazon, J.M.: \(C_0\)-semigroups norm continuous at infinity. Semigroup Forum 52(2), 213–224 (1996)
Nagel, R.J.: Mittelergodische Halbgruppen linearer operatoren. Ann. Inst. Fourier (Grenoble) 23(4), 75–87 (1973)
Acknowledgments
The author was supported by the Hungarian Research Fund (OTKA-100461).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Abdelaziz Rhandi.
To Professor Rainer Nagel on the occasion of his 75th birthday.
Rights and permissions
About this article
Cite this article
Farkas, B. A note on the periodic decomposition problem for semigroups. Semigroup Forum 92, 587–597 (2016). https://doi.org/10.1007/s00233-015-9719-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00233-015-9719-z