Some reproducing kernel spaces of continuous functions

By a result of L. Schwartz, a symmetric function is the reproducing kernel of a reproducing kernel Krein space if and only if it can be written as a difference of two positive functions; it seems, in general, difficult to check this last criteria. In the present study we show that a n × n valued symmetric function K(t, s) of class b³ for t, s ε{lunate} (a, b) is the reproducing kernel of a reproducing kernel Krein space of continuous functions. We first obtain a more general result when the symmetry hypothesis is removed and the Krein space is replaced by a pair of Hilbert spaces in duality with respect to a sesquilinear form.