## Abstract

We consider the family of arithmetical matrices given explicitly by

$$

E=\left\{\frac{[n,m]^t}{(nm)^{(\rho+t)/2}}\right\}_{n,m=1}^\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 $\ell^2(\bbN)$ 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{\varkappa}{n^{\rho-t}}(1+o(1)), \quad n\to\infty,

$$

with some $\varkappa>0$ and the negative eigenvalues obey the same relation, with the same asymptotic coefficient $\varkappa$.

We also indicate a connection of the spectral analysis of $E$ to the theory of Beurling primes.

$$

E=\left\{\frac{[n,m]^t}{(nm)^{(\rho+t)/2}}\right\}_{n,m=1}^\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 $\ell^2(\bbN)$ 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{\varkappa}{n^{\rho-t}}(1+o(1)), \quad n\to\infty,

$$

with some $\varkappa>0$ and the negative eigenvalues obey the same relation, with the same asymptotic coefficient $\varkappa$.

We also indicate a connection of the spectral analysis of $E$ to the theory of Beurling primes.

Original language | English |
---|---|

Journal | Pure and Applied Functional Analysis |

Publication status | Accepted/In press - 6 Jan 2024 |