Invariant subspaces for pairs of projections

A simple geometrical argument shows that every pair of projections on a fin ite-dimensional complex vector space has a common invariant subspace of dim ension 1 or 2. The idea extends to certain pairs of projections on an infin ite-dimensional Hilbert space H. In particular every projection on H has a reducing subspace, although a finite-dimensional one need not exist. In a f inal section, the results are extended to the existence of hyperinvariant s ubspaces for pairs of projections.