Three results in linear dynamics

In this article, first we show that the Fréchet space $H(\mathbb{D})$ cannot support strongly supercyclic weighted composition operators. Then we compute the constant $\epsilon$ for weighted backward shifts on ${\mathrm{\ell }}^p$ ($1\le p<\mathrm{\infty }$) and $c_0$. This constant is used to find strongly hypercyclic scalar multiples of non-invertible strongly supercyclic Banach space operators. Finally, we give an affirmative answer to a recent open question concerning supercyclic vectors.