Reducibility of self-adjoint linear relations and application to generalized Nevanlinna functions

• Date: 2022
• Main Author: Borogovac, M.
• Format: Article
• Language: English
• Published: Institute of Mathematics, NAS of Ukraine 2022
• Online Access: https://umj.imath.kiev.ua/index.php/umj/article/view/6084
• Journal Title: Ukrains’kyi Matematychnyi Zhurnal

UDC 517.9We present necessary and sufficient conditions for the reducibility of a self-adjoint linear relation in a Krein space. Then a generalized Nevanlinna function $Q$ represented by a self-adjoint linear relation $A$ in a Pontryagin space is decomposed by means of the reducing subspaces of $A.$ The sum of two functions $Q_{i}{\in N}_{\kappa_{i}}(\mathcal{H}),$ $i=1, 2,$ minimally represented by the triplets $(\mathcal{K}_{i},A_{i},\Gamma_{i})$ is also studied. For this purpose, we create a model $(\tilde{\mathcal{K}},\tilde{A},\tilde{\Gamma })$ to represent $Q:=Q_{1}+Q_{2}$ in terms of $(\mathcal{K}_{i},A_{i},\Gamma_{i})$. By using this model, necessary and sufficient conditions for $\kappa =\kappa_{1}+\kappa_{2}$ are proved in the analytic form. Finally, we explain how degenerate Jordan chains of the representing relation $A$ affect the reducing subspaces of $A$ and the decomposition of the corresponding function $Q.$