## Abstract

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 L_{1} ⊂ E, containing L as an infinite-codimensional subspace, such that L_{1} ∩ R = {0} and cl T(L_{1}) = 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 H_{0} in H satisfying H_{0} ∩ R = {0} and codim(H_{0} + R) = ∞ there exists an orthogonal decomposition H = V ⊕_{⊥} W such that V contains H_{0} 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.

Original language | English |
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Pages (from-to) | 203-216 |

Number of pages | 14 |

Journal | Studia Mathematica |

Volume | 246 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jan 2019 |

## Keywords

- Operator range
- Quasicomplemented subspace
- Separable Banach space

## ASJC Scopus subject areas

- General Mathematics