Characterization of Weyl Functions in the Class of Operator-Valued Generalized Nevanlinna Functions - Borogovac, Muhamed (Sarajevo Journal of Mathematics, 2024)
We provide the necessary and sufficient conditions for a generalized Nevanlinna function Q (QN_κ ( H )) to be a Weyl function (also known as a Weyl-Titchmarch function). We also investigate an important subclass of N_κ (H), the functions that have a boundedly invertible derivative at infinity Q'( ∞ ):=_z → ∞zQ(z). These functions are regular and have the operator representation Q( z )=Γ^+( A-z )^-1Γ,zρ ( 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 A, where A is the representing relation of Q:=-Q^-1. We illustrate our results through examples, wherein we begin with a given function QN_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. 2020 Mathematics Subject Classification. 34B20, 47B50, 47A06, 47A56