In this paper we prove a quaternionic positive real lemma as well as its generalized version, in case the associated kernel has negative squares for slice hyperholomorphic functions. We consider the case of functions with positive real part in the half space of quaternions with positive real part, as well as the case of (generalized) Schur functions in the open unit ball.
Alpay, D., Colombo, F., Lewkowicz, I. et al. Integr. Equ. Oper. Theory (2019) 91: 4. https://doi.org/10.1007/s00020-019-2503-9