Determinantal point processes with J-Hermitian correlation kernels

Let X be a locally compact Polish space and let m be a reference Radon measure on X. Let .X denote the configuration space over X, that is, the space of all locally finite subsets of X. A point process on X is a probability measure on .X. A point process . is called determinantal if its correlation functions have the form k(n)(x1,.,xn)=det[K(xi,xj)]i,j=1,.,n. The function K(x,y) is called the correlation kernel of the determinantal point process .. Assume that the space X is split into two parts: X=X1.X2. A kernel K(x,y) is called J-Hermitian if it is Hermitian on X1.X1 and X2.X2, and K(x,y)=..................K(y,x) for x.X1 and y.X2. We derive a necessary and sufficient condition of existence of a determinantal point process with a J-Hermitian correlation kernel K(x,y).