Somewhere dense Cesàro orbits and rotations of Cesàro hypercyclic operators

George Costakis¹ ; Demetris Hadjiloucas²

¹ Department of Mathematics University of Crete Knossos Avenue GR-714 09 Heraklion, Crete, Greece
² Department of Applied Mathematics University of Crete Knossos Avenue GR-714 09 Heraklion, Crete, Greece

Studia Mathematica, Tome 175 (2006) no. 3, pp. 249-269

Résumé

Let $T$ be a continuous linear operator acting on a Banach space $X$. We examine whether certain fundamental results for hypercyclic operators are still valid in the Cesàro hypercyclicity setting. In particular, in connection with the somewhere dense orbit theorem of Bourdon and Feldman, we show that if for some vector ${x\in X}$ the set $\{ Tx, \frac{T^2}{2}x, \frac{T^3}{3}x,\ldots \}$ is somewhere dense then for every $0\varepsilon 1$ the set $(0,\varepsilon ) \{Tx, \frac{T^2}{2}x, \frac{T^3}{3}x,\ldots \}$ is dense in $X$. Inspired by a result of Feldman, we also prove that if the sequence $\{ n^{-1} T^n x \}$ is $d$-dense then the operator $T$ is Cesàro hypercyclic. Finally, following the work of León-Saavedra and Müller, we consider rotations of Cesàro hypercyclic operators and we establish that in certain cases, for any $\lambda$ with $|\lambda |=1$, $T$ and $\lambda T$ share the same sets of Cesàro hypercyclic vectors.

Détail

DOI : 10.4064/sm175-3-4

Keywords: continuous linear operator acting banach space examine whether certain fundamental results hypercyclic operators still valid ces hypercyclicity setting particular connection somewhere dense orbit theorem bourdon feldman vector set frac frac ldots somewhere dense every varepsilon set varepsilon frac frac ldots dense inspired result feldman prove sequence d dense operator ces hypercyclic finally following work n saavedra ller consider rotations ces hypercyclic operators establish certain cases lambda lambda lambda share sets ces hypercyclic vectors