The Hilbert $L$-matrix

We analyze spectral properties of the Hilbert $L$-matrix $\left(\frac{1}{\max(m,n)+ν}\right)_{m,n=0}^{\infty}$ regarded as an operator $L_ν$ acting on $\ell^{2}(\mathbb{N}_{0})$, for $ν\in\mathbb{R}$, $ν\neq0,-1,-2,\dots$. The approach is based on a spectral analysis of the inverse of $L_ν$, 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 $L_ν$ 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.

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