An extension theorem for a sequence of Krein space contractions (Q1751319)

Summary: Consider Krein spaces \(\mathcal{U}\) and \(\mathcal{Y}\) and let \(\mathcal{H}_k\) and \(\mathcal{K}_k\) be regular subspaces of \(\mathcal{U}\) and \(\mathcal{Y}\), respectively, such that \(\mathcal{H}_k \subset \mathcal{H}_{k + 1}\) and \(\mathcal{K}_k \subset \mathcal{K}_{k + 1}\) (\(k \in \mathbb{N}\)). For each \(k \in \mathbb{N}\), let \(A_k : \mathcal{H}_k \rightarrow \mathcal{K}_k\) be a contraction. We derive necessary and sufficient conditions for the existence of a contraction \(B : \mathcal{U} \rightarrow \mathcal{Y}\) such that \(B|_{\mathcal{H}_k} = A_k\). Some interesting results are proved along the way.