Invertible elements in a convolution algebra of holomorphic functions
Let \(f(z)=\sum_{\nu=0}^ \infty f_ \nu z^ \nu\) and \(g(z)=\sum_{\nu=0}^ \infty g_ \nu z^ \nu\) be power series. Then the Hadamard product (or convolution) of \(f\) and \(g\) is defined by \[ (f*g)(z):=\sum_{\nu=0}^ \infty f_ \nu g_ \nu z^ \nu. \] For a region \(G\subset\widehat\mathbb{C}\) let \(H(G)\) denote the vector space of all functions being holomorphic in \(G\). If \(G\) with \(0\in G\) is such that the power series \(f*g\) defines a function in \(H(G)\) for all \(f,g\in H(G)\), then \(H(G)\) with \(*\) as multiplication turns out to be a commutative complex algebra. Furthermore, if \(1\not\in G\), then \(H(G)\) has an identity, namely the geometric series \(\gamma(z)=\sum_{\nu=0}^ \infty z^ \nu\). Some examples of regions \(G\) as above are \(G=\mathbb{D}\) (unit disk), \(G=\mathbb{C}\setminus[1,\infty)\), \(G=\widehat \mathbb{C}\setminus \{z_ 1,\dots,z_ k\}\) or \(G=\mathbb{C}\setminus \{z_ 1,\dots, z_ k\}\), where \(z_ 1,\dots,z_ k\) denote the \(k\)th roots of unity for some \(k\in\mathbb{N}\). One aim of this paper is to determine the invertible elements of the algebra \(H(G)\) for certain regions \(G\). If \(f\in H(G)\) being invertible it is necessary that \(f_ \nu\neq 0\) for all \(\nu\in\mathbb{N}_ 0\) and \(\lim_{\nu\to\infty} | f_ \nu|^{1/\nu}=1\). In the case of the unit disk these conditions are also sufficient. One conclusion of the results in this paper is that \(f\in H(\widehat C\setminus\{1\})\) is invertible, if and only if \(f(z)=c\gamma(z)+d\) with constants \(c,d\in\mathbb{C}\) such that \(c\neq 0\) and \(c+d\neq 0\). For \(H(\mathbb{C} \setminus[1,\infty))\) the invertible elements are characterized in terms of interpolatory functions \(\varphi\) of exponential type in the right half plane (i.e., \(\varphi(\nu)=f_ \nu\)). Though the investigation of invertible elements was the starting point of this work, the results actually go beyond this question.
scientific article; zbMATH DE number 86209