Duhamel algebras and applications
type: Item
instance of: scholarly article
title: Duhamel algebras and applications
zbMATH Open document ID: 1408.46048
publication date: 2018-06-25T00:00:00Z
review text: The author calls a Duhamel algebra a unital Banach algebra of functions in \(\mathrm{Hol}(\mathbb D)\) whose multiplication is given by the Duhamel product \((f\odot g)(z)= \int_0^z f'(z-t)g(t) \,dt + f(0)g(z)\). It was shown by \textit{N. M. Wigley} that this product defines an algebra structure in \(\mathrm{Hol}(\mathbb D)\) [Duke Math. J. 41, 211--217 (1974; Zbl 0283.30036)] and also in \(H^p(\mathbb D)\) for \(1\leq p\leq \infty\) [Can. Math. Bull. 18, 597--603 (1975; Zbl 0324.46051)]. Recently, this product was also analysed for Bergman spaces [\textit{H. Guediri} et al., New York J. Math. 21, 339--350 (2015; Zbl 1329.47034)]. In this paper, the author deals with general spaces of analytic functions \(X \subset\mathrm{Hol}(\mathbb D)\) where the integration operator \(Jf(z)=\int_0^z f(t)\,dt\) is bounded on \(X\), \(\{z^n\}\) defines a complete uniformly minimal system and satisfies \(\sum_{n,m\geq i+2} \frac{n!m!}{(n-m-i)!}\frac{\|z^{n+m-i}\|}{\|z^n\|\|z^m\|}<\infty\) for all \(i\geq 0\). Under such assumptions, the author shows that \((X, \odot)\) becomes a unital Duhamel algebra and its nontrivial closed ideals are given by \(\mathrm{span} \{z^{i+n}: i=0,1,\dots\}\) for some \(n\in \mathbb N\). Some applications of this result are provided.
reviewed by: Oscar Blasco
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 46E15
zbMATH DE Number: 6893830
zbMATH Keywords: Duhamel algebra
zbMATH Keywords: closed ideal
zbMATH Keywords: Hardy space
zbMATH Keywords: Banach space of analytic functions
zbMATH Keywords: invariant subspace
author: Mubariz T. Karaev
MaRDI profile type: Publication
full work available at URL: s10688-018-0201-z
OpenAlex ID: W2795250184
cites work: Volterra invariant subspaces of \(H^p\)
cites work: Q4800211
cites work: On the spectral analysis of direct sums of Riemann-Liouville operators in Sobolev spaces of vector functions
cites work: The Bergman space as a Banach algebra
cites work: Best constants for the Riesz projection
cites work: Q5467489
cites work: Some applications of Banach algebra techniques
cites work: On extended eigenvalues and extended eigenvectors of some operator classes
cites work: Q3455515
cites work: Q4380681
cites work: Q4871484
cites work: A local algebra structure for $H^p$ of the polydisc
cites work: The Duhamel product of analytic functions
cites work: A Banach Algebra Structure for Hp
DOI: 10.1007/S10688-018-0201-Z
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 6893830