Reproducing kernel Krein spaces of non-commutative functions

Laurent Schwartz showed that a hermitian kernel K is the difference of two positive kernels iff it has an associated reproducing kernel Krein space of functions. We generalize this equivalence to the free non-commutative setting of nc functions\kernels, with possible application to the locally bounded (equivalently, uniformly nc analytic) case (generalizing a result of Alpay in the classical commutative setting).