Hankel operators and the Stieltjes moment problem
type: Item
instance of: scholarly article
title: Hankel operators and the Stieltjes moment problem
zbMATH Open document ID: 1183.47022
author: Hélène Bommier-Hato
author: El-Hassan Youssfi
published in: Journal of Functional Analysis
publication date: 2010-01-11T00:00:00Z
review text: Let \(s\) be a non-vanishing Stieltjes moment sequence and let \(\mu\) be a representing measure of it. By \(\mu_n\) denote the image measure in \({\mathbb C}^n\) of \(\mu\otimes\sigma_n\) under the map \((t,\xi)\mapsto\sqrt{t}\xi\), where \(\sigma_n\) is the rotation-invariant probability measure on the unit sphere. By \({\mathcal A}(s)\) denote the space of holomorphic functions \(f(z)=\sum_{\alpha\in{\mathbb N}_0^n}a_\alpha z^\alpha\) satisfying \(\sum_{\alpha\in{\mathbb N}_ 0^n}{\alpha! s_{|\alpha|}\over (|\alpha|+n-1)!}|a_\alpha|^2<+\infty\). It is shown that this set is a reproducing kernel Hilbert space. On this space, consider the Hankel operator \(H_f\) with symbol \(f\). For a polynomial \(f\), let \(\text{Spec}(f)\) be the set of multi-indices \(k\in{\mathbb N}_ 0^n\) such that \({\partial^k f\over\partial z^k}(0)\neq 0\). One of the main results of the paper says that, if \(f\) is a holomorphic polynomial, then {\parindent=6mm \begin{itemize}\item[(a)] \(H_{\overline{f}}\) is bounded if and only if \[ \sup_{d\in {\mathbb N}_0}\left({s_{d+2|k|}\over s_{d+|k|}}-{s_{d+|k|}\over s_d}+{n-1\over d}{s_{d+|k|}\over s_d}\right)<+\infty \] for \(k\in\text{Spec}(f)\); \item[(b)] \(H_{\overline{f}}\) is compact if and only if \[ \lim_{d\to +\infty}\left({s_{d+2|k|}\over s_{d+|k|}}-{s_{d+|k|}\over s_d}+{n-1\over d}{s_{d+|k|}\over s_d}\right)=0 \] for \(k\in\text{Spec}(f)\); \item[(c)] If \(p>0\), then \(H_{\overline{f}}\) belongs to the Schatten class \(S_p\) if and only if \[ \sum_{d\in{\mathbb N}}d^{n-1}\left({s_{d+2|k|}\over s_{d+|k|}}-{s_{d+|k|}\over s_d}\right)^{p/2}+(n-1)d^{n-1-p/2}\left({s_{d+|k|}\over s_d}\right)^{p/2}<+\infty \] for all \(k\in\text{Spec}(f)\). \end{itemize}} In particular, it is shown that, if \(n=1\), then there are nontrivial Hilbert-Schmidt Hankel operators with anti-holomorphic symbols if and only if \(s\) is exponentially bounded. In this case, the space of symbols of such operators is the classical Dirichlet space.
reviewed by: Oleksiy Karlovych
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 47A57
zbMATH DE Number: 5655705
zbMATH Keywords: Stieltjes moment problem
zbMATH Keywords: Hankel operator
zbMATH Keywords: boundeness
zbMATH Keywords: compactness
zbMATH Keywords: Schatten class
zbMATH Keywords: Hilbert-Schmidt operator
zbMATH Keywords: anti-holomorphic symbol
zbMATH Keywords: reproducing kernel Hilbert space
zbMATH Keywords: Fock space
zbMATH Keywords: Bergman kernel
MaRDI profile type: Publication
full work available at URL: j.jfa.2009.08.004
OpenAlex ID: W1987131932
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DOI: 10.1016/J.JFA.2009.08.004
description: scientific article; zbMATH DE number 5655705