On the linear inverse from right operator for the convolution operator on the spaces of germs of analytical functions on convex compacts in \(\mathbb{C}\)
type: Item
instance of: scholarly article
title: On the linear inverse from right operator for the convolution operator on the spaces of germs of analytical functions on convex compacts in \(\mathbb{C}\)
zbMATH Open document ID: 0906.46019
author: Z. Momm
author: Sergej N. Melikhov
published in: Russian Mathematics
publication date: 1998-10-18T00:00:00Z
review text: Let \(G\) and \(K\) be convex compact subsets of the complex plane, and \(A(G)\) and \(A(G+ K)\) denote the spaces of germs of analytic functions on \(G\) and \(G+K\), resp. Every analytic functional \(\mu\) determined by \(K\) generates a linear continuous convolution operator \(T_\mu: A(G+K)\to A(G)\) by \(T_\mu(g)(z)= \mu(g(\cdot+ z))\), \(g\in A(G+ K)\). The problem of existence of a linear continuous right inverse operator to \(T_\mu\) is under consideration. Let \(G\) differ from a point, \(T_\mu\) be surjective, and let the zero set of the function \(\mu(\exp(\cdot z))\) be infinite. Under these assumptions, two necessary and sufficient conditions for existence of a linear continuous right inverse to \(T_\mu\) are obtained in terms of properties of the support function of the set \(G\).
Mathematics Subject Classification ID: 46E10
Mathematics Subject Classification ID: 44A35
Mathematics Subject Classification ID: 46F15
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 47A05
zbMATH DE Number: 1175202
zbMATH Keywords: spaces of germs of analytic functions
zbMATH Keywords: analytic functional
zbMATH Keywords: linear continuous convolution operator
zbMATH Keywords: linear continuous right inverse operator
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1175202