Abstract.
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.
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Alpay, D., Shapiro, M. Reproducing Kernel Quaternionic Pontryagin Spaces. Integr. equ. oper. theory 50, 431–476 (2004). https://doi.org/10.1007/s00020-003-1230-3
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DOI: https://doi.org/10.1007/s00020-003-1230-3