Some Remarks on Reproducing Kernel Krein Spaces
The one-to-one correspondence between positive functions and reproducing kernel Hilbert spaces was extended by L. Schwartz to a (onto, but not one-to-one) correspondence between difference of positive functions and reproducing kernel Krein spaces. After discussing this result, we prove that matrix value function K(z,ω) symmetric and jointly analytic in z and ω in a neighborhood of the origin is the reproducing kernel of a reproducing kernel Krein space. We conclude with an example showing that such a function can be the reproducing kernel of two different Krein spaces.
D. Alpay, Some remarks on reproducing kernel Krein spaces, The Rocky Mountain Journal of Mathematics, vol. 21 (1991) 1189-1205.
Rocky Mountain Mathematics Consortium