Reproducing Kernels and Riccati Equations

The purpose of this article is to present a brief exposition of the role of Riccati equations in the theory of reproducing kernel spaces. In particular, we shall exhibit a connection between positive semidefinite solutions of matrix Riccati equations and a class of finite dimensional reproducing kernel Hilbert spaces of rational vector valued functions, and an analogous (but more general) connection between selfadjoint solutions of matrix Riccati equations and finite dimensional reproducing kernel Krein spaces of rational vector valued functions. The reproducing kernels of the former are expressed in terms of a rational J inner matrix valued function Θ(λ) (see formula (15) below), whereas the reproducing kernels of the latter are expressed in terms of the same formula, but now Θ(λ) is only J unitary on the boundary of the region of interest. A more comprehensive account of parts of this analysis will appear in (Dym, 2001). The paper is organized as follows: In Sections 2–4 we will review a number of concepts from the theory of reproducing kernel spaces. Much of this analysis is carried out in a general notation that permits one to develop the theory simultaneously for a general region Ω+ in the complex plane that can be taken equal to either the open unit disc , the open upper half plane + , or the open right half plane +. The symbol ρω(λ) is then defined by the rule