A class of operator equations in analytic spaces
type: Item
instance of: scholarly article
title: A class of operator equations in analytic spaces
zbMATH Open document ID: 0531.47027
DOI: 10.1007/BF01093098
author: N. Ye. Linchuk
author: S. S. Linchuk
published in: Ukrainian Mathematical Journal
publication date: 1983-01-01T00:00:00Z
review text: Let \(G_ j (j=1,2)\) be a domain in \({\mathbb{C}}\), \(A(G_ j)\) the space of all analytic functions on \(G_ j\), \(U_{\phi_ i}\) the multiplication operator \((\phi_ j\in A(G_ j))\). This paper deals with the operator equation \(TU_{\phi_ 1}=U_{\phi_ 2}T\) in the class of all linear continuous operators \(L(A(G_ 1);A(G_ 2))\). In order to get the main results the author uses the Köthe integral representation of T. Theorem 1. Let \(t(\lambda\),z) (where \(\lambda \in {\mathbb{C}}-G_ 1\), \(z\in G_ 2)\) be the characteristic function of the Köthe integral representation of T. Then T is a solution of the equation if and only if \((\phi_ 1(\lambda)-\phi_ 2(z))t(\lambda,z)\) has analytic continuation onto \(G_ 1\times G_ 2\). (Other theorems deal with this equation in some special cases.)
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 47A50
zbMATH DE Number: 3842548
zbMATH Keywords: Köthe integral representation
MaRDI profile type: Publication
cites work: Dualität in der Funktionentheorie.
cites work: Q3852708
cites work: Operators commuting with multiplication in spaces of analytic functions of one variable
description: scientific article; zbMATH DE number 3842548