## Taiwanese Journal of Mathematics

### SUPERCYCLIC AND CESÀRO HYPERCYCLIC WEIGHTED TRANSLATIONS ON GROUPS

Chung-Chuan Chen

#### Abstract

Let $G$ be a locally compact group and let $1 \leq p \lt \infty$. We characterize supercyclic weighted translation operators on the Lebesgue space $L^p(G)$ in terms of the weight. Using this result, the characterization for Cesàro hypercyclic weighted translation operators is given. We also determine when scalar multiples of weighted translation operators are hypercyclic and topologically mixing, and show, for any weighted translation operator $T$, $\beta T$ is mixing for all $\beta \in (1,4)$ if $T$ and $4T$ are mixing.

#### Article information

Source
Taiwanese J. Math., Volume 16, Number 5 (2012), 1815-1827.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406799

Digital Object Identifier
doi:10.11650/twjm/1500406799

Mathematical Reviews number (MathSciNet)
MR2970687

Zentralblatt MATH identifier
1275.47020

#### Citation

Chen, Chung-Chuan. SUPERCYCLIC AND CESÀRO HYPERCYCLIC WEIGHTED TRANSLATIONS ON GROUPS. Taiwanese J. Math. 16 (2012), no. 5, 1815--1827. doi:10.11650/twjm/1500406799. https://projecteuclid.org/euclid.twjm/1500406799

#### References

• C. Badea, S. Grivaux and V. Müller, Multiples of hypercyclic operators, Proc. Amer. Math. Soc., 137 (2009), 1397-1403.
• C. Chen, Chaotic weighted translations on groups , Arch. Math. 97 (2011), 61-68.
• C. Chen and C-H. Chu, Hypercyclicity of weighted convolution operators on homogeneous spaces, Proc. Amer. Math. Soc., 137 (2009), 2709-2718.
• C. Chen and C-H. Chu, Hypercyclic weighted translations on groups, Proc. Amer. Math. Soc., 139 (2011), 2839-2846.
• K.-G. Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. $($N.S.$),$, 36 (1999), 345-381.
• K.-G. Grosse-Erdmann, Recent developments in hypercyclicity, RACSAM Rev. R. Acad. Cien. Ser. A. Mat., 97 (2003), 273-286.
• S. Grosser and M. Moskowitz, On central topological groups, Trans. Amer. Math. Soc., 127 (1967), 317-340.
• E. Hewitt and K. A. Ross, Abstract harmonic analysis, Springer-Verlag, Heidelberg, 1979.
• T. Kalmes, Hypercyclic, mixing, and chaotic $C_0$-semigroups induced by semiflows, Ergodic Theory Dynam. Systems, 27 (2007), 1599-1631.
• F. León-Saavedra, Operators with hypercyclic Cesàro means, Studia Math., 152 (2002), 201-215.
• F. León-Saavedra and V. Müller, Rotations of hypercyclic and supercyclic operators, Integral Equ. Oper. Theory, 50 (2004), 385-391.
• A. Montes-Rodríguez and H. Salas, Supercyclic subspaces: spectral theory and weighted shiftes, Adv. Math., 163 (2001), 74-134.
• H. Salas, Hypercyclic weighted shifts, Trans. Amer. Math. Soc., 347 (1995), 993-1004.
• H. Salas, Supercyclicity and weighted shifts, Studia Math., 135 (1999), 55-74.