An Operator Associated with De Branges Spaces and Universality Limits

Under suitable conditions on a measure, universality limits f ( ; ) that arise in the bulk, unitary case, are reproducing kernels of de Branges spaces of entire functions. In the classical case, f is the sinc kernel f (s; t) = sin (s t) (s t) ; but other kernels can arise. We study the linear operator L [h] (x) = Z 1 1 f (s; x)h (s) ds; establishing inequalities, and deducing some conditions for f to equal the sinc kernel. 1. Introduction and Results Let be a …nite positive Borel measure on R with all moments R xjd (x), j 0, …nite, and with in…nitely many points in its support. Then we may de…ne orthonormal polynomials pn (x) = nx n + :::; n > 0; n = 0; 1; 2; ::: satisfying the orthonormality conditions Z pnpmd = mn: Throughout we use 0 (x) = d dx to denote the almost everywhere existing Radon-Nikodym derivative of : Orthogonal polynomials play an important role in random matrix theory, especially in the unitary case [2], [4], [17]. One of the key limits there involves the reproducing kernel (1.1) Kn (x; y) = n 1 X k=0 pk (x) pk (y) : Because of the Christo¤el-Darboux formula, it may also be expressed as (1.2) Kn (x; y) = n 1 n pn (x) pn 1 (y) pn 1 (x) pn (y) x y ; x 6= y: Date : August 30, 2009. Research supported by NSF grant DMS0400446 and US-Israel BSF grant 2004353 1