On the bounds for the norms of Cauchy-Toeplitz and Cauchy-Hankel matrices
The authors investigate lower and upper bounds for Cauchy-Toeplitz matrices \(T_n= \frac k{1+k(i-j)}\) and for Cauchy-Hankel matrices \(H_n=\frac 1{(1/k)+(i+j)}\), \(k\in {\mathbb N}\) in the spectral norm \(\|\cdot\|_2\), and for the Hadamar product of Cauchy-Toeplitz and Cauchy-Hankel matrices in the Euclidean norm. The first main theorem states that \({\|T_n\|=\frac \pi {\sin (\pi/k)}}\), \(k\in{\mathbb N}\), \(n\) arbitrary. However, this statement is false. At first, both \(T_n\) and the fraction in the right size are not defined for \(k=1\). Secondly, it is known that the number \(\frac \pi {\sin (\pi/k)}\) is transcendental for any \(k\in{\mathbb N}\). On the other hand, the spectral norm of the matrix \(T_n\) with rational elements (\(=\max \sqrt {|\lambda (T_nT_n^*)|}\)) is an algebraic number. Therefore \(\|T_n\|_2\) cannot be equal to \(\frac \pi {\sin (\pi/k)}\) for any \(n\in {\mathbb N}\).
scientific article; zbMATH DE number 1861235