Reproducing kernel quaternionic pontryagin spaces

This paper studies various aspects of reproducing kernel spaces with a possibly indefinite metric when the field of scalar is replaced by the skew-field of quaternions. We first discuss in some details the positive case. A key fact which allows to consider the non-positive case is that Hermitian matrices with quaternionic entries have only real eigenvalues. This permits to extend the notion of functions with a finite number of negative squares to the present setting and we prove in particular that there is a one-to-one correspondence between such functions and reproducing kernel Pontryagin quaternionic spaces.