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Redundant decompositions, angles between subspaces and oblique projections

G. Corach and A. Maestripieri

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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.

Article information

Publ. Mat., Volume 54, Number 2 (2010), 461-484.

First available in Project Euclid: 28 June 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46C05: Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) 47A62: Equations involving linear operators, with operator unknowns 94A12: Signal theory (characterization, reconstruction, filtering, etc.) 41A65: Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)

Oblique projections angles between subspaces compatibility abstract splines


Corach, G.; Maestripieri, A. Redundant decompositions, angles between subspaces and oblique projections. Publ. Mat. 54 (2010), no. 2, 461--484.

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