Angular operators of nondegenerate subspaces (Q1814359)

The angular operator is usually attached to a positively semi-defined maximal linear subspace of a Krein space \((H,[.,.])\). In this paper the author extends this construction and some characteristic properties to an arbitrary nondegenerate subspace. Such a generalization is based on the fact that for each nondegenerate subspace \(L\) there exists a unique symmetry, namely \({\mathcal I}=\text{sgn}(P+JPJ-I)\), where \(J\) is the fundamental symmetry and \(P\) is the orthogonal projector on \(L\), such that \(L\) is maximal \({\mathcal I}\)-positive and its \({\mathcal I}\)-orthogonal complement is \(L^{[\perp]}\). The main results establish that for each nondegenerate \(L\) there exists a unique decomposition \(H=F\oplus\tilde F\) and a pure contraction \(K: F\to\tilde F\) such that \(L=\{f+Kf:\;f\in F\}\) and \(F^{[\perp]}=\{\tilde f+K^*\tilde f:\;\tilde f\in\tilde F\}\). In addition \(L\) is orthocomplemented iff \(\| K\| <1\). The degeneration of \(L\) is expressed in terms of satisfying the Riccati equation \(XBX+XA+CX+B=0\) by \(X=\pm K\), where \(J={A\;B \choose B^*\;C}\).