A note on non-supercyclic vectors of Hilbert space operators

In this note it is shown that there is a bounded linear operator $T$ on the Hardy Hilbert space $H^2$ and a vector $f$ in $H^2$ such that the closure of the set $\{\alpha T^{n}f:\alpha \in ℂ,n\ge 0\}$ is not $H^2$, but for every subsequence ${\left(n_k\right)}_{k=1}^{\infty }$ the closed linear span of $\{T^{n_k}f:k\ge 1\}$ is the whole space $H^2$. Furthermore, the closure of $\{T^{n}g:n\ge 0\}$ is $H^2$ for some $g\in H^2$.