A class of operator equations in analytic spaces
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.)
scientific article; zbMATH DE number 3842548