1. Introduction

We present here the main results of the theory of reproducing kernel Pontryagin spaces [Schwartz 1964; Sorjonen 1975] including some recent improvements [Alpay et al. 1997]. The paper is expository and is intended for nonspecialists in the indefinite theory. We presume knowledge of the Hubert space case, that is, Aronszajn's theory [1950] of reproducing kernel Hubert spaces. The main point is that much of the experience with the Hubert space theory is transferable to Pontryagin spaces. Section 2 presents background from operator theory. A key result here is a theorem to construct contraction operators by specifying their action on dense sets; we give a new proof that reduces the result to the isometric case. The main results on reproducing kernels are in Section 3.

Scalar-valued functions are assumed throughout. See [Alpay et al. 1997] for the extension to vector-valued functions and a detailed account of the theory of generalized Schur functions and associated colligations and reproducing kernel Pontryagin spaces.

2. Contraction operators on Pontryagin spaces

Inner products are assumed to be linear and symmetric and defined on a complex vector space. Orthogonality and direct sum are defined for any inner product space as in the Hubert space case.