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

Abstract

Necessary and sufficient conditions for reducidibility of a self-adjoint linear relation in a Krein space are given. Then a generalized Nevanlinna function $Q$, represented by a self-adjoint linear relation $A$, is decomposed by means of the reducing subspaces of $A$. The sum of two functions $Q_i{\in N}_{{\kappa }_i}\left(\mathcal{H}\right),i=1,2$, minimally represented by the triplets $\left({\mathcal{K}}_i,A_i,{\Gamma }_i\right)$, is also studied. For that purpose, a model $(\tilde{\mathcal{K}},\tilde{A},\tilde{\Gamma })$ to represent $Q:=Q_1+Q_2$ in terms of $\left({\mathcal{K}}_i,A_i,{\Gamma }_i\right)$ is created. By means of that model, necessary and sufficient conditions for $\kappa ={\kappa }_1+{\kappa }_2$ are proven in analytic terms. At the end, it is explained how degenerate Jordan chains of the representing relation $A$ affect reducing subspaces of $A$ and decomposition of the corresponding function $Q$.

27 pages; accepted in Ukr. Math. J

References in corpus (1)

• Inverse of generalized Nevanlinna function that is holomorphic at infinity

Cited by in corpus (1)

• Characterization of Weyl functions in the class of operator-valued generalized Nevanlinna functions