Reproducing kernel kreĭn spaces

Abstract

This chapter is an introduction to reproducing kernel Kre?in spaces and their interplay with operator valued Hermitian kernels. Existence and uniqueness properties are carefully reviewed. The approach used in this survey involves the more abstract, but very useful, concept of linearization or Kolmogorov decomposition, as well as the underlying concepts of Kre?in space induced by a selfadjoint operator and that of Kre?in space continuously embedded. The operator range feature of reproducing kernel spaces is emphasized. A careful presentation of Hermitian kernels on complex regions that point out a universality property of the Szegö kernels with respect to reproducing kernel Kre?in spaces of holomorphic functions is included.