Polar decomposition of oblique projections
type: Item
instance of: scholarly article
title: Polar decomposition of oblique projections
zbMATH Open document ID: 1193.47002
published in: Linear Algebra and its Applications
publication date: 2010-06-22T00:00:00Z
review text: Let \(\mathcal H\) be a Hilbert space and \(L(\mathcal H)\) the algebra of all bounded linear operators on \(\mathcal H\). The (canonical) polar decomposition of \(T \in L(\mathcal H)\) is the unique factorization \(T=V_{T}A_{T}\), where \(V_{T}\) is a partial isometry, \(A_{T} \in L(\mathcal H)^{+}\), the cone of all positive semidefinite operators on \(\mathcal H\), and \(N(V_{T})=N(A_{T})\) (here, \(N\) denotes the nullspace). Denoting by \(\mathcal Q\) the set of all oblique projectors, i.e., idempotents on \(\mathcal H\), and by \(\mathcal I\) the set of all partial isometries on \(\mathcal H\), the authors characterize the sets \(\mathcal I_{\mathcal Q}\)=\(\{V \in \mathcal I:\) there exists \(Q \in \mathcal Q \) such that \(V=V_{Q} \}\) and \(L(\mathcal H) ^{+}_{\mathcal Q}\)=\(\{A \in L(\mathcal H) ^{+}:\) there exists \(Q \in \mathcal Q \) such that \(A=A_{Q} \}\). The main results of the paper are the characterizations \[ \mathcal I_{ \mathcal Q } = \{ V \in \mathcal I : VP_{R(V)} \in L(\mathcal H)^{+},\;R(VP_{R(V)})=R(V)\} \] and \[ L(\mathcal H) ^{+}_{\mathcal Q} = \{ A \in L( \mathcal H )^{+}: \gamma(A)\geq 1,\;\dim \overline{ R(A-P_{R(A)})} \leq \dim N(A) \}, \] where \(P_{\mathcal S}\) denotes the orthogonal projection onto a subspace \(\mathcal S\) and \(\gamma(A)=\inf \{\|A\xi\|: \xi \in N(A)^{\bot}\), \(\|\xi\| = 1 \}\) is the reduced minimal modulus of \(A\). It is also proved that the mapping \(Q \mapsto V_{Q}\) is continuous and injective with inverse \(V \mapsto (V^{2}V^{*})^{\dagger}V\), and for each \(A \in L(\mathcal H)^{+}\) the set \(\{Q \in \mathcal Q: |Q| = A \}\) is characterized.
reviewed by: Edward L. Pekarev
Mathematics Subject Classification ID: 47A05
zbMATH DE Number: 5724596
zbMATH Keywords: oblique projections
zbMATH Keywords: polar decomposition
zbMATH Keywords: partial isometries
zbMATH Keywords: Moore-Penrose pseudoinverse
author: Gustavo Corach
author: A. Maestripieri
describes a project that uses: mftoolbox
MaRDI profile type: Publication
full work available at URL: j.laa.2010.03.016
OpenAlex ID: W2083262570
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DOI: 10.1016/J.LAA.2010.03.016
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description: scientific article; zbMATH DE number 5724596