Given a positive definite, bounded linear operator A on the Hilbert space H0 := l 2(E), we consider a reproducing kernel Hilbert space H+, H- with kernel functions A(x, y) , B(x, y). We investigate the ratios of determinants of some partial matrices of A and B. We also get a variational principle on the limit ratios of these values .We apply this relation to show the Giianness of the determinantal point process defined by the operator A(I + A) ?1 on the set E.
