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 Sκ and every nonempty open subset Ω of the unit disk D, there exist points z1, . . . , zn ∈ Ω such that the n × n Pick matrix h 1−f(zi)f(zj)∗ 1−ziz̄j in j,i=1 has κ negative eigenvalues. In this paper we discuss existence of an integer n0 such that any Pick matrix based on z1, . . . , zn ∈ Ω with n ≥ n0 has κ negative eigenvalues. Definitely, the answer depends on Ω. We prove that if Ω = D, 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 D then such a number n0 exists for every function f in Sκ. Mathematics Subject Classification (2000). 41A05, 32A35.