CONICET | Buscador de Institutos y Recursos Humanos

Let $cH$ be a complex Hilbert space. We study the relationships between the angles between closed subspacesof $cH$, the oblique projections associated to non direct decompositions of $cH$ and a notion of compatibility between a positive (semidefinite) operator $A$ acting on $cH$ and a closed subspace $cS$ of $cH$. It turns out that the compatibility is ruled by the values of the Dixmier angle between the orthogonal complement $cS^perp$ of $cS$ and the closure of $AcS$. We show that every redundant decomposition $cH=cS+cM^perp$(where redundant means that $cScapcM^perp$ 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.