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arXiv · 2411.07324

The Hilbert matrix done right

A. Montes-Rodríguez·J. A. Virtanen
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Abstract

We give very simple proofs of the classical results of Magnus and Hill on the spectral properties of the Hilbert matrix $$ H = \left ( {1 \over i+j+ 1 } \right )_{i,j\geq 0} $$ which defines a bounded linear operator on the sequence space $\ell^2$. In particular, we use the Mehler-Fock transform to find the spectrum and the latent eigenfunctions of the Hilbert matrix, that is, we show that the spectrum of $H$ is $[0,\pi]$ with no eigenvalues (Magnus' result) and describe all complex sequences $x$ such that $Hx=\mu x$ for some complex number $\mu$ (Hill's result).

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A. Montes-Rodríguez, J. A. Virtanen. 2024-11-11. The Hilbert matrix done right. https://arxiv.org/abs/2411.07324

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