This question comes out of my interest for positive definite kernels over the natural numbers. (I have collected some kernels with proofs).
First let me point to a connection between self-adjoint operators $O$ on $H:=l_2(\mathbb{N})$ and positive definite kernels $k$ defined on the natural numbers.
Let $H:=l_2(\mathbb{N})$ be the Hilbert space of sequences over the complex numbers. Suppose that we are given an operator $O=R^*R$, where $R$ is a linear bounded operator. Then $O$ is self adjoint, and denoting the standard orthonormal basis $(e_n)_{n \in \mathbb{N}}$ and $\phi(n):= R(e_n)$, we get:
$$\left<O(e_m),e_n\right> = \left<R^*R(e_m),e_n \right> = \left<R(e_m),R(e_n)\right> = \left < \phi(m),\phi(n) \right> =: k(m,n)$$
The left hand side denotes the entry of an infinte matrix $(\left <O(e_m),e_n \right>) = (k(m,n))$. The right hand side denotes a positive definite kernel $k$ on the natural numbers.
Probably the more interesting case is the other direction: Given a positive definite kernel $k$ on the natural numbers (such as $k(a,b) = \gcd(a,b), \min(a,b), \frac{\min(a,b)}{\max(a,b)}, \frac{\gcd(a,b)}{ab}$), then by the Moore-Aronszajn theorem, there exists a feature mapping:
$$\phi: \mathbb{N} \rightarrow F$$
such that:
$$\left< \phi(a), \phi(b) \right>_F = k(a,b)$$
Now define $R$ through: $R(e_n) := \phi(n)$, and let $O = R^* R$. Then $O$ is one self-adjoint operator associated with the positive definite kernel $k$ over the natural numbers, and we have again the infinite matrix $(k(i,j))_{1 \le i,j \le \infty}$ corresponding to $O$.
I think this is the reason, why the positive definite kernels over the natural numbers seem to pop up everywhere, for example in the formulation of the abc-conjecture:
$$K(a,b) := \frac{2(a+b)}{\gcd(a,b)\operatorname{rad}\left ( \frac{ab(a+b)}{\gcd(a,b)^3}\right )^2} $$
Another example where they show up, are elliptic curves:
Questions:
Suppose that $O$ is one / the operator in the Hilbert-Polya conjecture. How does the Riemann hypothesis follow from the existence of $O$? (I found the answer to this question here: https://www.redalyc.org/journal/5117/511766757001/html/ )
Is the spectrum of $O$ (independent of the Hilbert-Polya conjecture) given by a limit in $N$ where spec(O_N), N->infinity, where $O_N = (k(m,n))_{1\le m,n\le N}$ is a Gram matrix. In symbols:
$$\operatorname{spec}(O) =^? \lim_{N->\infty} \operatorname{spec}(O_N) $$?
