  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 $A(x,y)$, $x,y\in E$, 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 $\mathcal{H}_-\supset\mathcal{H}_0\supset\mathcal{H}_+$
becomes a rigged Hilbert space. We investigate a relationship between the
ratios of determinants of some partial matrices related to $A$ and $B$ and the
suitable projections in $\mathcal{H}_-$ and $\mathcal{H}_+$. 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 $E$. It
turns out that the class of determinantal point processes that can be
recognized as Gibbs measures for suitable interactions is much bigger than that
obtained by Shirai and Takahashi.
