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

Abstract

There is no supercyclic power bounded operator of class $C_{1\cdot }.$ 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 $C_{1\cdot }$, then it has an extension $\hat{T}$ which is a weakly l-sequentially supercyclic singular-continuous unitary (and $\hat{T}$ has a Rajchman scalar spectral measure whenever $T$ is weakly stable)$.$ The above result implies ${\sigma }_P(T)={\sigma }_P(T^{\ast })=\varnothing$, and also that if a weakly l-sequentially supercyclic operator is similar to an isometry, then it is similar to a unitary operator.

References in corpus (1)

• Asymptotic limits, Banach limits, and Cesàro means