Abstract
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.
Citation
G. Corach. A. Maestripieri. "Redundant decompositions, angles between subspaces and oblique projections." Publ. Mat. 54 (2) 461 - 484, 2010.
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