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 subset of E such that L boolean AND R = {0} and codim(L + R) = infinity, we provide a condition to ensure the existence of an infinite-dimensional closed subspace L-1 subset of E, containing L as an infinite-codimensional subspace, such that L-1 boolean AND 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 boolean AND R = {0} and codim(H-0 + R) = infinity there exists an orthogonal decomposition H = V circle plus(perpendicular to) W such that V contains H-0 as an infinite-codimensional subspace and V boolean AND R = W boolean AND R = {0}. We also obtain generalizations of some classical results on quasicomplemented subspaces of Banach spaces.
Operator ranges and quasicomplemented subspaces of Banach spaces / V.P. Fonf, S. Lajara, S. Troyanski, C. Zanco. - In: STUDIA MATHEMATICA. - ISSN 0039-3223. - 246:2(2019), pp. 203-216. [10.4064/sm180110-31-1]
Operator ranges and quasicomplemented subspaces of Banach spaces
C. Zanco
Ultimo
2019
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 subset of E such that L boolean AND R = {0} and codim(L + R) = infinity, we provide a condition to ensure the existence of an infinite-dimensional closed subspace L-1 subset of E, containing L as an infinite-codimensional subspace, such that L-1 boolean AND 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 boolean AND R = {0} and codim(H-0 + R) = infinity there exists an orthogonal decomposition H = V circle plus(perpendicular to) W such that V contains H-0 as an infinite-codimensional subspace and V boolean AND R = W boolean AND R = {0}. We also obtain generalizations of some classical results on quasicomplemented subspaces of Banach spaces.Pubblicazioni consigliate
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