The Hilbert $L$-matrix

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 $\nu \in \mathbb{R}$, $\nu \ne 0,-1,-2,\dots$. 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.

31 pages, 6 figures, accepted for publication in the Journal of Functional Analysis