Hilbert matrix on Bergman spaces
type: Item
instance of: scholarly article
title: Hilbert matrix on Bergman spaces
zbMATH Open document ID: 1080.47031
published in: Illinois Journal of Mathematics
publication date: 2005-03-01T00:00:00Z
review text: The Hilbert matrix \(H\) is defined as an infinite matrix with entries \(a_{i,j}=(i+j+1)^{-1}\), where \(i\), \(j\geq 0\). A classical inequality due to Hilbert allows us to determine the norm of \(H\) viewed as an operator acting on the standard sequence spaces \(l^{p}\). Since the space \(l^ 2\) can be identified with the classical Hardy space \(H^ 2\) and there is a similar way of viewing the Bergman space \(A^ 2\) of all square-integrable analytic functions on the unit disk in terms of its Taylor coefficients, this motivates analogous questions about the norm of \(H\) on the general Hardy spaces \(H^ p\) of the disk and on the Bergman spaces \(A^ p\) of analytic functions in the unit disk which are \(p\)-integrable with respect to area measure. The purpose of this paper is to study the norm of the operator \(H\) induced by the Hilbert matrix as an operator on \(A^{p}\). The action of the operator can be seen in the following way: it transforms the function whose sequence of Taylor coefficients is \((a_{n})_{n=0}^\infty\) into the function with Taylor coefficients \((\sum_{k=0}^\infty {a_{k}\over n+k+1})_{n=0}^\infty\). In the author's earlier paper with \textit{A. G. Siskakis} [Stud. Math. 140, No. 2, 191--198 (2000; Zbl 0980.47029)], an upper bound for the norm of this operator was obtained on the Hardy spaces \(H^{p}\), \(p>1\). The author shows that \(H\) is also bounded on \(A^ p\) when \(p>2\), but cannot possibly be bounded on \(A^ 2\) (using a duality argument). The question whether this operator is actually well defined on \(A^ 2\) is left open; it appears, though, that the answer is negative. The main result of the paper gives two upper bounds for the norm of \(H\) on \(A^ p\). The one obtained in the case \(4\leq p<\infty\) is analogous to the Hardy space result: \(\| H\| \leq \pi/\sin (2\pi/p)\), while a different one is obtained in the case \(2<p<4\). The method of proof relies on the representation of the Hilbert matrix in terms of a weighted compos…
reviewed by: Dragan Vukotić
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 47A30
Mathematics Subject Classification ID: 32A36
Mathematics Subject Classification ID: 47B35
zbMATH DE Number: 2140267
zbMATH Keywords: Hilbert matrix
zbMATH Keywords: Bergman spaces
zbMATH Keywords: coefficient multiplication
zbMATH Keywords: weighted composition operators
zbMATH Keywords: operator norm
author: Epaminondas A. Diamantopoulos
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2140267