Operators with closed ranges in spaces of analytic vector-valued functions
Suppose X and Y are Banach spaces and \(z\to T_ z\) is an analytic function, defined in some open neighbourhood G of a compact polydisk \(P\subset {\mathbb{C}}\) with values in the space \(L(X,Y)\) of bounded linear operators from X into Y. Let \(A_+(P,X)\) denote the Banach space of X- valued analytic functions with absolutely convergent Taylor series. The author studies the ''multiplication operator'' \(\tilde T:\) \(A_+(P,X)\to A_+(P,Y)\) defined by \((\tilde Tf)(z)=T_ z(f(z))\). His main result states that \(\tilde T\) has a closed range in case the ranges of all the \(T_ z\) are closed and depend continuously on z, more precisely: if there is a Banach space Z and an analytic function S from G into \(L(Y,Z)\) such that \(Im\;T_ z=\ker S_ z\) for all \(z\in P\), then \[ Im \tilde T=\{g\in A^+(P,Y): g(z)\in Im\;T_ z\text{ for all} z\in P\}. \] As an application of his main theorem, the author constructs Banach spaces of sections of some infinite dimensional analytic sheaves.
Banach spaces of sections of some infinite dimensional analytic sheaves
scientific article; zbMATH DE number 3987992