This is a question I was thinking of some time ago.
Suppose $\mathbf{X} \equiv (X, \|\cdot\|_X)$ is a (real or complex) Banach space, $U$ is a dense subspace of $\mathbf{X}$, and $\phi$ is a bounded linear operator $(U,\|\cdot\|_X) \to \mathbf{X}$. We know from the B.L.T. theorem that $\phi$ can be uniquely extended to a bounded linear operator $\Phi: \mathbf{X} \to \mathbf{X}$.
Question. Provided $x \in X$, does there exist a sequence, $\{x_n\}_{n=1}^\infty$, in $U$ such that $\lim_n x_n = x$ in $\mathbf{X}$ and, for each $n \in \mathbb{N}^+$, $\{\Phi^k(x_n)\}_{k=1}^n \subseteq \phi(U)$?