A Variational Principle in the Dual Pair of Reproducing Kernel Hilbert Spaces and an Application

Abstract

Given a positive definite, bounded linear operator A on the Hilbert space $$\mathcal{H}$$ 0≔l 2(E), we consider a reproducing kernel Hilbert space $$\mathcal{H}$$ + with a reproducing kernel A(x,y). Here E is any countable set and A(x,y), x,y∊ E, is the representation of A w.r.t. the usual basis of $$\mathcal{H}$$ 0. Imposing further conditions on the operator A, we also consider another reproducing kernel Hilbert space $$\mathcal{H}$$ − with a kernel function B(x,y), which is the representation of the inverse of A in a sense, so that $$\mathcal{H}$$ −⊃ $$\mathcal{H}$$ 0⊃ $$\mathcal{H}$$ + becomes a rigged Hilbert space. 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 Gibbsianness of the determinantal point process (or fermion point process) defined by the operator A(I+A)−1 on the set E.