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

There is no supercyclic power bounded operator of class $C_{1\cdot }\mathrm{.}$ There exist, however, weakly l-sequentially supercyclic unitary operators$\mathrm{.}$ We show that if $T$ is a weakly l-sequentially supercyclic power bounded operator of class $C_{1\cdot }$, 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)$\mathrm{.}$ The above result implies ${\sigma }_P(T)={\sigma }_P(T^{\ast })=\mathrm{\varnothing }$, and also that if a weakly l-sequentially supercyclic operator is similar to an isometry, then it is similar to a unitary operator.