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

Abstract

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 ∈ N κ i ( ℋ ) , i = 1,2 , minimally represented by the triplets ( 𝒦 i , A i , Γ i ) is also studied. For this purpose, we create a model ( 𝒦 ˜ , A ˜ , Γ ˜ ) to represent Q : = Q 1 + Q 2 in terms of ( 𝒦 i , A i , Γ i ) . By using this model, necessary and sufficient conditions for κ = κ 1 + κ 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 .