On a theorem due to Murray

In this paper, we introduce the notions of α-quasicomplemented and totally α-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\ α-quasicomplemented if, and only if, α<₀ ( this is an analogue of the theorem of Murray-Mackey and Lindenstrauss). We also show that if H is a Hilbert space and Y,W⊂ H are closed subspaces of H such that W is orthogonal to Y and codim( Y+W) =∞, then Y has a quasicomplement Z containing W with Z/W=∞. Other results in the different contexts are also included. Such results establish a connection between the theory of quasicomplemented subspaces and ( α,β)-spaceability.