Spectral asymptotics for a family of LCM matrices

We consider the family of arithmetical matrices given explicitly by $$E(\sigma ,\tau )={\left\{\frac{n^{\sigma }{m}^{\sigma }}{[n,m{]}^{\tau }}\right\}}_{n,m=1}^{\mathrm{\infty }},$$ where $[n,m]$ is the least common multiple of $n$ and $m$ and the real parameters $\sigma$ and $\tau$ satisfy $\rho :=\tau -2\sigma >0$, $\tau -\sigma >\frac{1}{2}$ and $\tau >0$. We prove that $E(\sigma ,\tau )$ is a compact self-adjoint positive definite operator on ${\mathrm{\ell }}^2(\mathbb{N})$, and the ordered sequence of eigenvalues of $E(\sigma ,\tau )$ obeys the asymptotic relation $${\lambda }_n(E(\sigma ,\tau ))=\frac{\mathrm{\varkappa }(\sigma ,\tau )}{n^{\rho }}+o(n^{-\rho }),n\to \mathrm{\infty },$$ with some $\mathrm{\varkappa }(\sigma ,\tau )>0$. We give an application of this fact to the asymptotics of singular values of truncated multiplicative Toeplitz matrices with the symbol given by the Riemann zeta function on the vertical line with abscissa $\sigma <1\mathrm{/}2$. We also point out a connection of the spectral analysis of $E(\sigma ,\tau )$ to the theory of generalised prime systems.