The injective hull of an operator ideal on locally convex spaces (Q788256)

If U is an operator ideal on locally convex spaces then the injective hull \(U^ J\) of U is defined to be the intersection of all injective ideals containing it. If E is a locally convex space we put \(E_{\infty}=\prod_{B}C(B)\), where \(\{\) \(B\}\) cover the family of all equicontinuous subsets of the dual of E and C(B) is the space of bounded scalarvalued functions on B. The authors show that if U is an operator ideal on locally convex spaces and E and F are locally convex then \(U^ J(E,F)\) consists exactly of the operators T:\(E\to F\) for which \(J_ FT\in U(E,F_{\infty})\), where \(J_ F\) is the natural embedding of F into \(F_{\infty}\). In case U is a bounded operator ideal the authors describe the injective hull of U using seminorm ideals. Similar constructions can be found in the book ''Operator Ideals'' (1978; Zbl 0399.47039) by \textit{A. Pietsch} for the operator ideals on Banach spaces.