On the spectrum of Hilbert matrix operator
Let \(l^p\) and \(H^p\), \(1<p<\infty\), denote the space of all \(p\)-summable sequences and the Hardy space, respectively. The author considers a generalization of the Hilbert matrix \[\mathcal{H}_\lambda=\left(\frac{1}{n+m+\lambda}\right)_{n,m=0}^{\infty},\;\; \lambda\not=0,-1,-2,\ldots,\] thought of as a bounded linear operator acting on \(l^p\) and on \(H^p\), \(1<p<\infty\). The author describes the essential spectrum of \(\mathcal{H}_\lambda\) on \(l^p\) and on \(H^p\) for \(\lambda\in\mathbb{C}\setminus\{0,-1,-2,\ldots\}\). For \(\lambda\in\mathbb{C}\setminus\mathbb{Z}\), the author describes the point spectrum of \(\mathcal{H}_\lambda\) acting on \(H^p\). This was done earlier in [\textit{A. Aleman} et al., Constr. Approx. 36, No. 3, 353--374 (2012; Zbl 1268.47040)], but here the author uses a different method. He then shows that the result extends to \(\lambda\in\{1,2,\ldots\}\). The author also determines the point spectrum of \(\mathcal{H}_\lambda\) acting on \(l^p\) for \(\lambda\in\mathbb{C}\setminus\{0,-1,-2,\ldots\}\).
scientific article; zbMATH DE number 7357789
scientific article