The Hilbert $L$-matrix

TL;DR

This paper investigates the spectral properties of the Hilbert L-matrix as an operator on , using hypergeometric functions to analyze its inverse, and resolves open questions about its operator norm.

Contribution

It provides a spectral analysis of the Hilbert L-matrix, including solutions to open problems on its operator norm and insights into its positivity and Fredholm determinants.

Findings

Determined the spectral properties of the Hilbert L-matrix.

Solved open problems regarding the operator norm of L_{ u}.

Discussed general aspects of L-operators, including positivity and Fredholm determinants.

Abstract

We analyze spectral properties of the Hilbert $L$-matrix $$\left(\frac{1}{\max(m,n)+\nu}\right)_{m,n=0}^{\infty}$$ regarded as an operator $L_{\nu}$ acting on $\ell^{2}(\mathbb{N}_{0})$, for $\nu\in\mathbb{R}$, $\nu\neq0,-1,-2,\dots$. The approach is based on a spectral analysis of the inverse of $L_{\nu}$, which is an unbounded Jacobi operator whose spectral properties are deducible in terms of the unit argument ${}_{3}F_{2}$-hypergeometric functions. In particular, we give answers to two open problems concerning the operator norm of $L_{\nu}$ published by L. Bouthat and J. Mashreghi in [Oper. Matrices 15, No. 1 (2021), 47--58]. In addition, several general aspects concerning the definition of an $L$-operator, its positivity, and Fredholm determinants are also discussed.