  A determinantal point process (DPP) is an ensemble of random
nonnegative-integer-valued Radon measures $\Xi$ on a space $S$ with measure
$\lambda$, whose correlation functions are all given by determinants specified
by an integral kernel $K$ called the correlation kernel. We consider a pair of
Hilbert spaces, $H_{\ell}, \ell=1,2$, which are assumed to be realized as
$L^2$-spaces, $L^2(S_{\ell}, \lambda_{\ell})$, $\ell=1,2$, and introduce a
bounded linear operator ${\cal W} : H_1 \to H_2$ and its adjoint ${\cal
W}^{\ast} : H_2 \to H_1$. We show that if ${\cal W}$ is a partial isometry of
locally Hilbert--Schmidt class, then we have a unique DPP on $(\Xi_1, K_1,
\lambda_1)$ associated with ${\cal W}^* {\cal W}$. In addition, if ${\cal W}^*$
is also of locally Hilbert--Schmidt class, then we have a unique pair of DPPs,
$(\Xi_{\ell}, K_{\ell}, \lambda_{\ell})$, $\ell=1,2$. We also give a practical
framework which makes ${\cal W}$ and ${\cal W}^{\ast}$ satisfy the above
conditions. Our framework to construct pairs of DPPs implies useful duality
relations between DPPs making pairs. For a correlation kernel of a given DPP
our formula can provide plural different expressions, which reveal different
aspects of the DPP. In order to demonstrate these advantages of our framework
as well as to show that the class of DPPs obtained by this method is large
enough to study universal structures in a variety of DPPs, we report plenty of
examples of DPPs in one-, two-, and higher-dimensional spaces $S$, where
several types of weak convergence from finite DPPs to infinite DPPs are given.
One-parameter ($d \in \mathbb{N}$) series of infinite DPPs on $S=\mathbb{R}^d$
and $\mathbb{C}^d$ are discussed, which we call the Euclidean and the
Heisenberg families of DPPs, respectively, following the terminologies of
Zelditch.
