A Szegő Limit Theorem Related to the Hilbert Matrix

The Szegő 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, $\alpha \ge \frac{1}{2}$, 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 $\mathrm{\mid }\beta \mathrm{\mid }<1$ to $\beta \in \mathbf{C}\setminus ]1,\mathrm{\infty }[$. The bound on the correction term is improved to $O(1)$ instead of $o(\ln (N))$ for $\beta \in \mathbf{C}\setminus [1,\mathrm{\infty }[$. The limit case $\beta =1$ is derived directly from the asymptotics for general $\beta$.