Reducibility of Self-Adjoint Linear Relations and Application to Generalized Nevanlinna Functions

摘要

We 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 can be decomposed by means of the reducing subspaces of A. The sum of two functions Q(i) is an element of N-ki (H), i = 1, 2, minimally represented by the triplets (K-i, A(i), gamma(i)) is also studied. For this purpose, we create a model ( (K), (A),(gamma)) to represent Q := Q(1) +Q(2) in terms of (K-i, A(i), gamma(i)). By using this model, necessary and sufficient conditions for k = k(1 )+ k(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.