A Szeg\H{o} Limit Theorem Related to the Hilbert Matrix

The Szeg\H{o} limit theorem by Fedele and Gebert for matrices of the type identity minus Hankel matrix is proved for the special case $1-\frac{\beta}{\pi}H_{N,\alpha}$ where $H_{N,\alpha}$ is the $N\times N$-Hilbert matrix, $\alpha\geq\frac{1}{2}$, and $\beta\in\mathbf{C}$. The proof uses operator theoretic tools and a reduction to the classical Kac--Akhiezer theorem for the Carleman operator. Thereby, the validity of the theorem for this special Hankel matrix can be extended from $|\beta|<1$ to $\beta\in\mathbf{C}\setminus ]1,\infty[$. The bound on the correction term is improved to $O(1)$ instead of $o(\ln(N))$ for $\beta\in\mathbf{C}\setminus [1,\infty[$. The limit case $\beta=1$ is derived directly from the asymptotics for general $\beta$.