Non-sequential weak supercyclicity and hypercyclicity

TL;DR

This paper investigates weak supercyclicity and hypercyclicity of operators on Banach spaces, constructing examples and establishing conditions under which these properties hold or differ, especially on Hilbert spaces and weighted bilateral shifts.

Contribution

It constructs a weakly supercyclic operator that is not weakly sequentially supercyclic, answering a question by Bayart and Matheron, and characterizes weakly supercyclic bilateral shifts on ll_p spaces.

Findings

Existence of weakly supercyclic operators not weakly sequentially supercyclic on Hilbert spaces.

Characterization of weakly supercyclic bilateral shifts on ll_p spaces.

Weakly hypercyclic bilateral shifts are norm hypercyclic for 1a0bca0p<2.

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.