Weakly Supercyclic Power Bounded Operators of Class C_{1.}

Abstract

There is no supercyclic power bounded operator of class $C_{1{\textstyle\cdot}}.$ There exist, however, weakly l-sequentially supercyclic unitary operators$.$ We show that if $T$ is a weakly l-sequentially supercyclic power bounded operator of class $C_{1{\textstyle\cdot}}$, then it has an extension $\widehat T$ which is a weakly l-sequentially supercyclic singular-continuous unitary (and $\widehat T$ has a Rajchman scalar spectral measure whenever $T$ is weakly stable)$.$ The above result implies $\sigma_{\kern-1ptP}(T)=\sigma_{\kern-1ptP}(T^*)=\varnothing$, and also that if a weakly l-sequentially supercyclic operator is similar to an isometry, then it is similar to a unitary operator.