Characterization of Weyl functions in the set of regular generalized Nevanlinna functions N κ ( H )
: Let Q ∈ N κ ( H ) be a regular function, minimally represented by a self-adjoint linear relation A in the Pontryagin space ( K , [ ., . ]) of negative index κ and let ˆ Q := − Q − 1 be represented by a linear relation ˆ A . We give a necessary and sufficient condition for Q to be a Weyl function associated with S = A ∩ ˆ A and A . We also study a class of functions Q ∈ N κ ( H ) that have boundedly invertible derivative at infinity Q ′ ( ∞ ) := lim z →∞ zQ ( z ); we give relation matrices of A , ˆ A and S + in terms of S . We prove that every such function Q is a Weyl function associated with S = A | I − P and A . In examples we show how to apply the main results. For instance, for a given regular function Q ∈ N κ ( H ) with boundedly invertible Q ′ ( ∞ ) represented by A , we find the symmetric relation S so that Q is the Weyl function associated with ( S, A ). Then we find the corresponding boundary triple Π = ( H , Γ 0 , Γ 1 ). In another example, we apply main results to find linear relations ˆ R , S , ˆ A , S + which are associated with a given regular function Q ∈ N κ ( H ) represented by a given operator A .