Monatshefte für Mathematik

, Volume 183, Issue 3, pp 397–413 | Cite as

Density of translates in weighted \({{\varvec{L}}}^{{\varvec{p}}}\) spaces on locally compact groups

  • Evgeny Abakumov
  • Yulia Kuznetsova


Let G be a locally compact group, and let \(1\leqslant p < \infty \). Consider the weighted \(L^p\)-space \(L^p(G,\omega )=\{f:\int |f\omega |^p<\infty \}\), where \(\omega :G\rightarrow \mathbb {R}\) is a positive measurable function. Under appropriate conditions on \(\omega \), G acts on \(L^p(G,\omega )\) by translations. When is this action hypercyclic, that is, there is a function in this space such that the set of all its translations is dense in \(L^p(G,\omega )\)? Salas (Trans Am Math Soc 347:993–1004, 1995) gave a criterion of hypercyclicity in the case \(G=\mathbb {Z}\). Under mild assumptions, we present a corresponding characterization for a general locally compact group G. Our results are obtained in a more general setting when the translations only by a subset \(S\subset G\) are considered.


Locally compact groups Weighted spaces Hypercyclicity Translation semigroups 

Mathematics Subject Classification

47A16 37C85 43A15 


  1. 1.
    Chen, C.-C.: Hypercyclic weighted translations generated by non-torsion elements. Arch. Math. (Basel) 101(2), 135–141 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Conejero, J.A., Peris, A.: Hypercyclic translation \(C_0\)-semigroups on complex sectors. Discrete Contin. Dyn. Syst. 25(4), 1195–1208 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    de Vries, J.: The local weight of an effective locally compact transformation group and the dimension of \(L^2(G)\). Colloq. Math. 39(2), 319–323 (1978)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Desch, W., Schappacher, W., Webb, G.F.: Hypercyclic and chaotic semigroups of linear operators. Ergodic Theory Dyn. Syst. 17(4), 793–819 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Edwards, R.E.: The stability of weighted Lebesgue spaces. Trans. Am. Math. Soc. 93, 369–394 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Feichtinger, H.G.: Gewichtsfunktionen auf lokalkompakten Gruppen. Sitzber. Österr. Akad. Wiss. Abt. II 188(8–10), 451–471 (1979)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Grosse-Erdmann, K.-G.: Universal families and hypercyclic operators. Bull. Am. Math. Soc. (N.S.) 36(3), 345–381 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Salas, H.: Hypercyclic weighted shifts. Trans. Am. Math. Soc. 347, 993–1004 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Tam, K.W.: On measures with separable orbit. Proc. Am. Math. Soc. 23, 409–411 (1969)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Wien 2017

Authors and Affiliations

  1. 1.University Paris-EstMarne-la-ValléeFrance
  2. 2.University of Bourgogne Franche-ComtéBesançonFrance

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