Spectral asymptotics for a family of arithmetical matrices and connection to Beurling primes

We consider the family of arithmetical matrices given explicitly by $$E={\left\{\frac{[n,m{]}^t}{(nm{)}^{(\rho +t)\mathrm{/}2}}\right\}}_{n,m=1}^{\mathrm{\infty }}$$ where $[n,m]$ is the least common multiple of $n$ and $m$ and the real parameters $\rho$ and $t$ satisfy $t>0$, $\rho >t+1$. We prove that $E$ is a compact self-adjoint operator on ${\mathrm{\ell }}^2(\mathbb{N})$ with infinitely many of both positive and negative eigenvalues. Furthermore, we prove that the ordered sequence of positive eigenvalues of $E$ obeys the asymptotic relation $${\lambda }_n^{+}(E)=\frac{\mathrm{\varkappa }}{n^{\rho -t}}(1+o(1)),n\to \mathrm{\infty },$$ with some $\mathrm{\varkappa }>0$ and the negative eigenvalues obey the same relation, with the same asymptotic coefficient $\mathrm{\varkappa }$. We also indicate a connection of the spectral analysis of $E$ to the theory of Beurling primes.