Polar decomposition of oblique projections
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.
scientific article; zbMATH DE number 5724596
scientific article