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∈ Nκ( 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'( ∞ ):=lim z → ∞zQ(z). These functions are regular and have the operator representation Q( z )=Γ̃⁺( A-z )⁻¹Γ̃,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:=Γ̃ ( Γ̃⁺ Γ̃ )⁻¹ Γ̃⁺. Additionally, we provide the relation matrices of the adjoint relation S⁺ of S, and of Â, where  is the representing relation of Q̂:=-Q⁻¹. We illustrate our results through examples, wherein we begin with a given function Q∈ Nκ( 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. 2020 Mathematics Subject Classification. 34B20, 47B50, 47A06, 47A56