Duhamel algebras and applications
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.
scientific article; zbMATH DE number 6893830