Redundant decompositions, angles between subspaces and oblique projections

Abstract

Let H be a complex Hilbert space. We study the relationships between the angles between closed subspaces of H, the oblique projections associated to non direct decompositions of H and a notion of compatibility between a positive (semidefinite) operator A acting on H and a closed subspace S of H. It turns out that the compatibility is ruled by the values of the Dixmier angle between the orthogonal complement S⊥ of S and the closure of AS. We show that every redundant decomposition H = S+M⊥ (where redundant means that S ∩M⊥ is not trivial) occurs in the presence of a certain compatibility. We also show applications of these results to some signal processing problems (consistent reconstruction) and to abstract splines problems which come from approximation theory.