A remark on a paper of J. Prada (Q1312358)

In this note we observe that any strictly singular operator from a Fréchet space \(E\) into a complete locally convex space \(F\) which admits a continuous norm must be bounded. Since a bounded strictly singular operator between Fréchet spaces is a Riesz-transformation, we obtain the following extended version of the result of Prada using the same methods as in [\textit{J. Prada}, this ARch. 43, 179-182 (1984; Zbl 0537.46005)] Theorem. Let the cartesian product \(E\times F\) of two countably normable Fréchet spaces be such that each continuous linear operator from \(E\) into \(F\) is super strictly singular. Then a complemented subspace of \(E\times F\) is isomorphic to a subspace of \(E\times F\) of the form \(E_ 0\times F_ 0\), where \(E_ 0\), respectively \(F_ 0\), is a complemented subspace of \(E\), respectively \(F\). The strictly singular and the super strictly singular operators between normed spaces coincide. So our result also generalizes the result of I. S. Edelstein and P. Woitaszczyk [\textit{J. Lindenstrauß} and \textit{L. Tzafini}, `Classical Banach spaces. I' (1977; Zbl 0362.46013)] about complemented subspaces of the direct sum of two Banach spaces.