The Hilbert matrix done right

Abstract

We give very simple proofs of the classical results of Magnus and Hill on the spectral properties of the Hilbert matrix $$H={\left(\frac{1}{i+j+1}\right)}_{i,j\ge 0}$$ 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 $[0,\ddot{I}€]$ with no eigenvalues (Magnus' result) and describe all complex sequences $x$ such that $Hx=\hat{I}¼x$ for some complex number $\hat{I}¼$ (Hill's result).

To appear in Operator Theory: Advances and Applications (IWOTA 2023)