An image problem for compact operators

Abstract

Let X \mathcal {X} be a separable Banach space and ( X n ) n (\mathcal {X}_n)_n a sequence of closed subspaces of X \mathcal {X} satisfying X n ⊂ X n + 1 \mathcal {X}_n\subset \mathcal {X}_{n+1} for all n n . We first prove the existence of a dense-range and injective compact operator K K such that each K X n K\mathcal {X}_n is a dense subset of X n \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).