On a theorem due to Murray

Abstract

In this paper, we introduce the notions of $\alpha$-quasicomplemented and totally $\alpha$-quasicomplemented subspaces and we established some results under these contexts. We show, for example, that if $X$ is a separable or reflexive Banach space and $Y$ is a closed infinite codimensional subspace of $X$, then $Y$ is totally$\mathit{\ }\alpha$-quasicomplemented if, and only if, $\alpha<\aleph_{0}$ $\left( \text{this is an analogue of the theorem of Murray-Mackey and Lindenstrauss}\right) $. We also show that if $H$ is a Hilbert space and $Y,W\subset H$ are closed subspaces of $H$ such that $W$ is orthogonal to $Y$ and $\operatorname{codim}\left( Y+W\right) =\infty$, then $Y$ has a quasicomplement $Z$ containing $W$ with $\dim Z/W=\infty$. Other results in the different contexts are also included. Such results establish a connection between the theory of quasicomplemented subspaces and $\left( \alpha,\beta\right) $-spaceability.