Reproducing kernel almost Pontryagin spaces

Abstract

An almost Pontryagin space A is an inner product space which admits a direct and orthogonal decomposition of the form A=A>[+˙]A≤ with a Hilbert space A> and a finite-dimensional negative semidefinite space A≤. A reproducing kernel almost Pontryagin space is an almost Pontryagin space of functions (defined on some nonempty set and taking values in some Krein space), with the property that all point evaluation functionals are continuous. We address two problems.1° In the presence of degeneracy, it is not possible to reproduce function values as inner products with a kernel function in the usual way. We obtain a natural substitute for a kernel function, and study the relation between spaces and kernel functions in detail.2° Given an inner product space L of functions, does there exist a reproducing kernel almost Pontryagin space A which contains L isometrically? We characterise those spaces for which the answer is “yes”. We show that, in case of existence, there is a unique such space A which contains L isometrically and densely. Its geometry, in particular its degree of degeneracy, is an important invariant of L. It plays a role in connection with Krein's formula describing generalised resolvents and, thus, in several concrete problems related with the extension theory of symmetric operators.