An image problem for compact operators

Let $\mathcal{X}$ be a separable Banach space and $(\mathcal{X}_n)_n$ a sequence of closed subspaces of $\mathcal{X}$ satisfying $\mathcal{X}_n\subset \mathcal{X}_{n+1}$ for all $n$. We first prove the existence of a dense-range and injective compact operator $K$ such that each $K\mathcal{X}_n$ is a dense subset of $\mathcal{X}_n$, solving a problem of Yahaghi (2004). Our second main result concerns isomorphic and dense-range injective compact mappings between dense sets of linearly independent vectors, extending a result of Grivaux (2003).