Asymptotic spectral properties of the Hilbert $L$-matrix

TL;DR

This paper investigates the asymptotic behavior of eigenvalues of the generalized Hilbert L-matrix for large sizes, providing new formulas for eigenvalue distributions and improving bounds related to Hardy's inequality.

Contribution

It derives the asymptotic distribution of eigenvalues for the generalized Hilbert L-matrix, including formulas for small and large eigenvalues, and refines bounds on the matrix norm.

Findings

Eigenvalues' asymptotic distribution outside the origin for general a9a0

Asymptotic formulas for small eigenvalues when a9>0

Improved asymptotic expansion of the matrix norm for a9=1

Abstract

We study asymptotic spectral properties of the generalized Hilbert $L$-matrix \[ L_{n}(\nu)=\left(\frac{1}{\max(i,j)+\nu}\right)_{i,j=0}^{n-1}, \] for large order $n$. First, for general $\nu\neq0,-1,-2,\dots$, we deduce the asymptotic distribution of eigenvalues of $L_{n}(\nu)$ outside the origin. Second, for $\nu>0$, asymptotic formulas for small eigenvalues of $L_{n}(\nu)$ are derived. Third, in the classical case $\nu=1$, we also prove asymptotic formulas for large eigenvalues of $L_{n}\equiv L_{n}(1)$. I particular, we obtain an asymptotic expansion of $\|L_{n}\|$ improving Wilf's formula for the best constant in truncated Hardy's inequality.