Construction of operators with prescribed orbits in Fréchet spaces with a continuous norm

Let $X$ be a separable, infinite--dimensional real or complex Fréchet space admitting a continuous norm. Let $\{v_n:\ n\geq 1\}$ be a dense set of linearly independent vectors of $X$. We show that there exists a continuous linear operator $T$ on $X$ such that the orbit of $v_1$ under $T$ is exactly the set $\{v_n:\ n\geq 1\}$. Thus, we extend a result of Grivaux for Banach spaces to the setting of non--normable Fr\'echet spaces with a continuous norm. We also provide some consequences of the main result.