Common complements of two subspaces of a Hilbert space

In this paper we find a necessary and sufficient condition for two closed subspaces, X and Y, of a Hilbert space H to have a common complement, i.e. a subspace Z having trivial intersection with X and Y and such that H=X+Z=Y+Z. Unlike the finite-dimensional case the condition is significantly more subtle than simple equalities of dimensions and codimensions, and non-trivial examples of subspaces without a common complement are possible.