CHARACTERIZATION OF WEYL FUNCTIONS IN THE CLASS OF OPERATOR-VALUED GENERALIZED NEVANLINNA FUNCTIONS.

We provide the necessary and sufficient conditions for a generalized Nevanlinna function Q (Q ∊ N_κ (H)) to be a Weyl function (also known as a Weyl-Titchmarch function). We also investigate an important subclass of Nκ(H), the functions that have a boundedly invertible derivative at infinity Q′ (∞):= limz→∞zQ(z). These functions are regular and have the operator representation Q(z)= ˜Γ+ (A-z)-1 ˜Γ, z ∊ ρ(A), where A is a bounded self-adjoint operator in a Pontryagin space K. We prove that every such strict function Q is a Weyl function associated with the symmetric operator S:= A|(I-P)K, where P is the orthogonal projection, P:= ˜Γ(˜Γ+ ˜Γ)^-1 ˜Γ+. Additionally, we provide the relation matrices of the adjoint relation S+ of S, and of &#194, where &#194 is the representing relation of Q:= -Q-1. We illustrate our results through examples, wherein we begin with a given function Q ∊ N_κ (H) and proceed to determine the closed symmetric linear relation S and the boundary triple Π so that Q becomes the Weyl function associated with Π. [ABSTRACT FROM AUTHOR]