Hypercyclicity of weighted translations on Orlicz spaces. (English) Zbl 1462.47004
Summary: In this paper, we study the hypercyclicity of the weighted translation \(C_{u,g}\) defined on Orlicz space \(L^\Phi(G)\) where \(G\) is a locally compact group, \(g\in G\) and \(u\) is a weight function on \(G\). It is shown that when \(g\in G\) is a torsion element, then \(C_{u,g}\) cannot be hypercyclic. However, for an aperiodic element \(g\in G\), necessary and sufficient conditions for \(C_{u,g}\) and its adjoint are given to be hypercyclic.
MSC:
| 47A16 | Cyclic vectors, hypercyclic and chaotic operators |
| 47B37 | Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
References:
| [1] | S. I. A NSARI, Hypercyclic and cyclic vectors, J. Funct. Anal. 128 (1995), no. 2, 374-383. · Zbl 0853.47013 |
| [2] | F. B AYART, ´E. M ATHERON, Dynamics of linear operators, Cambridge Tracts in Mathematics, Cam bridge University Press, Cambridge, 2009. · Zbl 1187.47001 |
| [3] | C. C HEN, C.-H. C HU, Hypercyclic weighted translations on groups, Proc. Amer. Math. Soc. 139 (2011), no. 8, 2839-2846. · Zbl 1221.47017 |
| [4] | C. C HEN, C.-H. C HU, Hypercyclicity of weighted convolution operators on homogeneous spaces, Proc. Amer. Math. Soc. 137 (2009), no. 8, 2709-2718. · Zbl 1177.47013 |
| [5] | K.-G. G ROSSE-E RDMANN, A. P. M ANGUILLOT, Linear chaos, Universitext, Springer, London, 2011. · Zbl 1246.47004 |
| [6] | C. K ITAI, Invariant closed sets for linear opeartors, Thesis (Ph. D.), University of Toronto (Canada), 1982. |
| [7] | M. M. R AO, Z. D. R EN, Theory of Orlicz spaces, Monographs and Textbooks in Pure and Applied Mathematics, 146, Marcel Dekker, Inc., New York, 1991. · Zbl 1096.30007 |
| [8] | H. N. S ALAS, Hypercyclic weighted shifts, Trans. Amer. Math. Soc., 347 (3) (1995) 993-1004. (Received February 18, 2017)M. R. Azimi Department of Mathematics Faculty of Sciences, University of Maragheh P.O. Box 55136-553, Maragheh, Iran e-mail:mhr.azimi@maragheh.ac.ir I. Akbarbaglu Department of Mathematics Farhangian University Tabriz, Iran e-mail:ibrahim.akbarbaglu@gmail.com Operators and Matrices www.ele-math.com |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
