caa a22 4500 467931852 CHVBK 20180406152945.0 cr unu---uuuuu 170328e20061101xx s 000 0 eng 10.1007/s00020-006-1428-2 doi (NATIONALLICENCE)springer-10.1007/s00020-006-1428-2 On Negative Inertia of Pick Matrices Associated with Generalized Schur Functions [Elektronische Daten] [Vladimir Bolotnikov, Alexander Kheifets] Abstract.: It is known [6] that for every function f in the generalized Schur class $$\mathcal{S}_{\kappa } $$ and every nonempty open subset Ω of the unit disk $$\mathbb{D}$$ , there exist points z1,...,zn ∈Ω such that the n × nPick matrix $${\left[ {\frac{{1 - f(z_{i} )f(z_{j} )^{*} }}{{1 - z_{i} \overline{z} _{j} }}} \right]}^{n}_{{j,i = 1}} $$ 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 $$\Omega = \mathbb{D}$$ , 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 $$\mathbb{D}$$ then such a number n0 exists for every function f in $$\mathcal{S}_{\kappa }$$ . Birkhäuser Verlag, Basel, 2006 Pick matrices nationallicence generalized Schur functions nationallicence Bolotnikov Vladimir Department of Mathematics, The College of William and Mary, 23187- 8795, Williamsburg, VA, USA aut Kheifets Alexander Department of Mathematics, University of Massachusetts, 01854, Lowell, MA, USA aut Integral Equations and Operator Theory Birkhäuser-Verlag; www.birkhäuser.ch 56/3(2006-11-01), 323-355 0378-620X 56:3<323 2006 56 20 https://doi.org/10.1007/s00020-006-1428-2 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00020-006-1428-2 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Bolotnikov Vladimir Department of Mathematics, The College of William and Mary, 23187- 8795, Williamsburg, VA, USA aut NATIONALLICENCE 700 1- Kheifets Alexander Department of Mathematics, University of Massachusetts, 01854, Lowell, MA, USA aut NATIONALLICENCE 773 0- Integral Equations and Operator Theory Birkhäuser-Verlag; www.birkhäuser.ch 56/3(2006-11-01), 323-355 0378-620X 56:3<323 2006 56 20 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer