TY - JOUR
AU - M. Borogovac
PY - 2022/08/09
Y2 - 2022/10/07
TI - Reducibility of self-adjoint linear relations and application to generalized Nevanlinna functions
JF - Ukrains’kyi Matematychnyi Zhurnal
JA - Ukr. Mat. Zhurn.
VL - 74
IS - 7
SE - Research articles
DO - 10.37863/umzh.v74i7.6084
UR - https://umj.imath.kiev.ua/index.php/umj/article/view/6084
AB - 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 $Q.$
ER -