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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
Article

Abstract

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.

Keywords

Locally compact groups Weighted spaces Hypercyclicity Translation semigroups 

Mathematics Subject Classification

47A16 37C85 43A15 

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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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