I am wondering about the connection between the kernel which gives the nth correlation function of a determinantal point process and the L^2 Hilbert space for which it uniquely defines an integral operator.
The definition of a determinantal point process defined from a kernel $K$ is as follows.
Let $\Lambda$ be a locally compact Polish space and $\mu$ be a Radon measure on $\Lambda .$ Also, consider a measurable function $K: \Lambda^{2} \rightarrow \mathbb{C}$. We say that $X$ is a determinantal point process on $\Lambda$ with kernel $K$ if it is a simple point process on $\Lambda$ with a joint intensity or correlation function (which is the density of its factorial moment measure) given by $$ \rho_{n}\left(x_{1}, \ldots, x_{n}\right)=\operatorname{det}\left[K\left(x_{i}, x_{j}\right)\right]_{1 \leq i, j \leq n} $$ for every $n \geq 1$ and $x_{1}, \ldots, x_{n} \in \Lambda .$
This definition however says nothing about the relationship between this kernel and the Hilbert space $L^2(\Lambda)$ for which it defines an integral operator by $T: L^2(\Lambda)\to L^2(\Lambda)$ by.
$$T(f)(x)=\int_\Lambda K(x,y) f(y)dy$$
What purpose does the Hilbert space of square-integrable functions $L^2(\Lambda)$ serve when working with determinantal point processes? Why might I be interested in such a bounded linear operator from $L^2(\Lambda)\to L^2(\Lambda)$?
It feels that the $L^2(\Lambda)$ is a separate object from the probability space and polish space over which the point processes is defined and to me, it is unclear why this is relevant.
The survey Soshnikov, Alexander. "Determinantal random point fields." Russian Mathematical Surveys 55, no. 5 (2000) is highly recommended and should clarify the matter for you.
We also tried our best to explain the connection in the book https://yuvalperes.com/zeros-of-gaussian-analytic-functions-and-determinantal-point-processes/ (there is a link to the PDF at that site.) In particular, if you look at Algorithm 4.2, you will see how the kernel can be used to sample from the point process. This was developed further in the book:
Kulesza, Alex, and Ben Taskar. "Determinantal point processes for machine learning." Foundations and Trends® in Machine Learning 5, no. 2–3 (2012): 123-286.