The Hilbert L-matrix

Abstract

We analyze spectral properties of the Hilbert L-matrix(1max⁡(m,n)+ν)m,n=0∞ regarded as an operator Lν acting on ℓ2(N0), for ν∈R, ν≠0,−1,−2,…. The approach is based on a spectral analysis of the inverse of Lν, which is an unbounded Jacobi operator whose spectral properties are deducible in terms of the unit argument F23-hypergeometric functions. In particular, we give answers to two open problems concerning the operator norm of Lν 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.