Redundant decompositions, angles between subspaces and oblique projections

Corach, G. ; Maestripieri, A.

Publ. Mat., Tome 54 (2010) no. 2, p. 461-484 / Harvested from Project Euclid

Résumé

Let ${\mathcal H}$ be a complex Hilbert space. We study the relationships between the angles between closed subspaces of ${\mathcal H}$, the oblique projections associated to non direct decompositions of ${\mathcal H}$ and a notion of compatibility between a positive (semidefinite) operator $A$ acting on ${\mathcal H}$ and a closed subspace ${\mathcal S}$ of ${\mathcal H}$. It turns out that the compatibility is ruled by the values of the Dixmier angle between the orthogonal complement ${\mathcal S}^\perp$ of ${\mathcal S}$ and the closure of $A{\mathcal S}$. We show that every redundant decomposition ${\mathcal H}={\mathcal S}+{\mathcal M}^\perp$ (where redundant means that ${\mathcal S}\cap{\mathcal M}^\perp$ 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.

Détail

Publié le : 2010-05-15
Classification:  Oblique projections,  angles between subspaces,  compatibility,  abstract splines,  46C05,  47A62,  94A12,  41A65