Operator ranges and quasicomplemented subspaces of Banach spaces
Volume 246 / 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 $L\subset E$ such that $L\cap R=\{0\}$ and $\mathop {\rm codim}\nolimits (L+R)=\infty $, we provide a condition to ensure the existence of an infinite-dimensional closed subspace $L_1\subset E$, containing $L$ as an infinite-codimensional subspace, such that $L_1\cap R= \{0\}$ and $\mathop {\rm cl}\nolimits T(L_1) = \mathop {\rm cl}\nolimits T(E)$. 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 $H_0\cap R = \{0\}$ and $\mathop {\rm codim}\nolimits (H_0+R)=\infty $ there exists an orthogonal decomposition $H=V\oplus _{\perp } W$ such that $V$ contains $H_0$ as an infinite-codimensional subspace and ${V \cap R = W \cap R =\{0\}}$. We also obtain generalizations of some classical results on quasicomplemented subspaces of Banach spaces.
