Somewhere dense Cesàro orbits and rotations of Cesàro hypercyclic operators
Volume 175 / 2006
Abstract
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.
