Publicacions Matemàtiques

Redundant decompositions, angles between subspaces and oblique projections

G. Corach and A. Maestripieri

Full-text: Open access

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.

Article information

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

Dates
First available in Project Euclid: 28 June 2010

Permanent link to this document
https://projecteuclid.org/euclid.pm/1277731542

Mathematical Reviews number (MathSciNet)
MR2675933

Zentralblatt MATH identifier
1204.46015

Subjects
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)

Keywords
Oblique projections angles between subspaces compatibility abstract splines

Citation

Corach, G.; Maestripieri, A. Redundant decompositions, angles between subspaces and oblique projections. Publ. Mat. 54 (2010), no. 2, 461--484. https://projecteuclid.org/euclid.pm/1277731542


Export citation