Operator ranges and quasicomplemented subspaces of Banach spaces

V. P. Fonf, S. Lajara, S. Troyanski, C. Zanco

Research output: Contribution to journalArticlepeer-review

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 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.

Original languageEnglish
Pages (from-to)203-216
Number of pages14
JournalStudia Mathematica
Volume246
Issue number2
DOIs
StatePublished - 1 Jan 2019

Keywords

  • Operator range
  • Quasicomplemented subspace
  • Separable Banach space

ASJC Scopus subject areas

  • Mathematics (all)

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