On Negative Inertia of Pick Matrices Associated with Generalized Schur Functions

It is known [6] that for every function f in the generalized Schur class and every nonempty open subset Ω of the unit disk , there exist points z₁,...,z_n ∈Ω such that the n × nPick matrix has κ negative eigenvalues. In this paper we discuss existence of an integer n₀ such that any Pick matrix based on z₁,...,z_n ∈Ω with n ≥ n₀ has κ negative eigenvalues. Definitely, the answer depends on Ω. We prove that if , then such a number n₀ does not exist unless f is a ratio of two finite Blaschke products; in the latter case the minimal value of n₀ can be found. We show also that if the closure of Ω is contained in then such a number n₀ exists for every function f in .