On the spectrum of Hilbert matrix operator
type: Item
instance of: scholarly article
title: On the spectrum of Hilbert matrix operator
zbMATH Open document ID: 1487.47050
published in: Integral Equations and Operator Theory
publication date: 2021-06-11T00:00:00Z
review text: 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 \(\lambda\in\mathbb{C}\setminus\{0,-1,-2,\ldots\}\).
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 47B33
Mathematics Subject Classification ID: 45E10
zbMATH DE Number: 7357789
zbMATH Keywords: Hilbert matrix operator
zbMATH Keywords: Toeplitz operator
zbMATH Keywords: Hankel operator
zbMATH Keywords: spectrum
zbMATH Keywords: essential spectrum
zbMATH Keywords: point spectrum
zbMATH Keywords: eigenfunction
zbMATH Keywords: hypergeometric function
author: B. Silbermann
MaRDI profile type: Publication
full work available at URL: s00020-021-02637-5
OpenAlex ID: W3157882632
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DOI: 10.1007/S00020-021-02637-5
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description: scientific article; zbMATH DE number 7357789