Reproducing Kernel in Krein Spaces

Abstract

This article describes a new form to introduce a reproducing kernel for a Krein space based on orthogonal projectors enabling to describe the kernel of a Krein space as the difference between the kernel of definite positive subspace and the kernel of definite negative subspace corresponding to kernel of the associated Hilbert space. As application, the authors obtain some basic properties of both kernels for Krein spaces and exhibit that each kernel is uniquely determined by the Krein space given. The methods and results employed generalize the notion of reproducing kernel given in Hilbert spaces to the context of spaces endowed with indefinite metric.