Operator ranges and quasicomplemented subspaces of Banach spaces

Given a bounded linear operator T from a separable infinite-dimensional Banach space E into a Banach space Y, an operator range R in E and a closed subspace L ⊂ E such that L ∩ R = {0} and codim (L + R) = ∞, we provide a condition to ensure the existence of an infinite-dimensional closed subspace L₁ ⊂ E, containing L as an infinite-codimensional subspace, such that L₁ ∩ R = {0} and cl T(L₁) = cl T(E). This condition enables us to build closed subspaces of E with a special behaviour with respect to an operator range in E. In particular, we show that if R is an operator range in a Hilbert space, then for every closed subspace H₀ in H satisfying H₀ ∩ R = {0} and codim(H₀ + R) = ∞ there exists an orthogonal decomposition H = V ⊕_⊥ W such that V contains H₀ as an infinite-codimensional subspace and V ∩ R = W ∩ R = {0}. We also obtain generalizations of some classical results on quasicomplemented subspaces of Banach spaces.