Non-sequential weak supercyclicity and hypercyclicity

Abstract

A bounded linear operator $T$ acting on a Banach space $\B$ is called weakly hypercyclic if there exists $x\in \B$ such that the orbit ${T^n x: n=0,1,...}$ is weakly dense in $\B$ and $T$ is called weakly supercyclic if there is $x\in \B$ for which the projective orbit ${\lambda T^n x: \lambda \in \C, n=0,1,...}$ is weakly dense in $\B$. If weak density is replaced by weak sequential density, then $T$ is said to be weakly sequentially hypercyclic or supercyclic respectively. It is shown that on a separable Hilbert space there are weakly supercyclic operators which are not weakly sequentially supercyclic. This is achieved by constructing a Borel probability measure $\mu$ on the unit circle for which the Fourier coefficients vanish at infinity and the multiplication operator $Mf(z)=zf(z)$ acting on $L_2(\mu)$ is weakly supercyclic. It is not weakly sequentially supercyclic, since the projective orbit under $M$ of each element in $L_2(\mu)$ is weakly sequentially closed. This answers a question posed by Bayart and Matheron. It is proved that the bilateral shift on $\ell_p(\Z)$, $1\leq p <\infty$, is weakly supercyclic if and only if $2<p<\infty$ and that any weakly supercyclic weighted bilateral shift on $\ell_p(\Z)$ for $1\leq p\leq 2$ is norm supercyclic. It is also shown that any weakly hypercyclic weighted bilateral shift on $\ell_p(\Z)$ for $1\leq p<2$ is norm hypercyclic, which answers a question of Chan and Sanders.