Krein-Langer factorizations via pole triples (Q1409585)

Let \({\mathcal G}, {\mathcal H}\) be Hilbert spaces, let \({\mathcal L}({\mathcal G},{\mathcal H})\) denote the space of continuous linear operators from \({\mathcal G}\) to \({\mathcal H}\), and let \(\mathbb D\) denote the open unit disc on \(\mathbb C\). An operator-valued function \(F: D_F \to {\mathcal L}({\mathcal G},{\mathcal H})\) with \(D_F \subseteq \mathbb C\) is called {meromorphic} on \(\mathbb D\) if for each \(z_0 \in \mathbb D\) there is an \(\varepsilon > 0\) such that \(D(z_0, \varepsilon) : = \{z \in \mathbb D: 0 < | z - z_0| < \varepsilon \} \subseteq D_F\) and \(F(z) = \sum_{j=-p}^{\infty} (z - z_0)^j F_j\) for some nonnegative integer \(p = p(z_0)\), some \(F_j = F_j(z_0) \in {\mathcal L}({\mathcal G},{\mathcal H})\) and each \(z \in D(z_0, \varepsilon)\). The {generalized Schur class} \(S_k({\mathcal G},{\mathcal H})\) is defined for a nonnegative integer \(k\) as follows: a (meromorphic on \(\mathbb D\)) \({\mathcal L}({\mathcal G},{\mathcal H})\)-valued function \(F \in S_k({\mathcal G},{\mathcal H})\) provided that the kernel \[ K_F(z,w) = \frac{I_{\mathcal H} - F(z) F(w)^*}{1 - z w^*} \] has \(k\) negative squares on \(D_F\). The main result (Theorem 4.1) asserts that for a function \(F\) the condition \(F \in S_k({\mathcal G},{\mathcal H})\) is valid if and only if \(F\) admits a so-called right (equivalently, left) Krein-Langer factorization. Besides, the authors characterize those triples of operators \((A,B,C)\) for which there are (analytic on \(\mathbb D\)) \({\mathcal L}({\mathcal G},{\mathcal H})\)-valued functions \(H\) with \(C (zI - A)^{-1}B + H(z) \in S_k({\mathcal G},{\mathcal H})\).