Examples of summing, integral and nuclear operators on the space \(C([0,1],X)\) with values in \(C_{0}\)
Let \(\Omega\) be a compact Hausdorff space, \(X,Y\) be Banach spaces and \(C(\Omega,X)\) be the space of continuous \(X\)-valued functions on \(\Omega\) under the uniform norm. When \(X\) is the scalar field, denote \(C(\Omega)\) instead of \(C(\Omega,X)\). If \(U:C(\Omega,X)\rightarrow Y\) is a bounded linear operator, define \(U^{\#}:C(\Omega)\rightarrow L(X,Y)\) by \((U^{\#} \varphi)(x)=U(\varphi\otimes x)\) for all \(\varphi\in C(\Omega)\) and \(x\in X\), where \((\varphi\otimes x)(w)=\varphi(w)x\), \(w\in\Omega\). The author obtains necessary and sufficient conditions for some operators \(U:C(\Omega,X)\rightarrow c_{0}\) to satisfy the hypothesis of some theorems (Swartz's theorem, Saab's theorem, among others) concerning absolutely summing, integral and nuclear operators, and involving \(U\) and \(U^{\#}\). Besides presenting a bulk of examples, in some cases the author also discusses whether the converses of these theorems are true or not. For example, it is presented a new counterexample for the converse of a theorem on absolutely summing operators due to \textit{S.\,Montgomery--Smith} and \textit{P.\,Saab} [Proc.\ R.\ Soc.\ Edinb., Sect.\,A 120, No.\,3/4, 283--296 (1992; Zbl 0793.47017)].
scientific article; zbMATH DE number 5146711
scientific article