Unbounded linear operators and nuclear Köthe quotients
A locally convex space F is said to satisfy condition (y) if there is a neighborhood \(V_ 1\) such that every \(u\in F'\) can be expressed as the \(\sigma (F',F)\)-limit of an equicontinuous net \((u_{\alpha})\) where each \(u_{\alpha}\in F'[V^ 0_ 1]\). In particular a Fréchet space F which has the bounded approximation property and admits a continuous norm satisfies (y). The main result of this note is that if there is an unbounded continuous operator T mapping a Fréchet space E into a Fréchet space F which satisfies (y), then there is a nuclear Köthe space \(\lambda\) (A) and a continuous surjection Q: \(F\to \lambda (A)\) such that \(QT(E)=\lambda (A).\) \textit{S. F. Bellenot} and \textit{E. Dubinsky} [Trans. Am. Math. Soc. 273, 579-594 (1982; Zbl 0494.46001)] proved that a separable Fréchet space F has a nuclear Köthe quotient if and only if it satisfies the following condition: there exists k such that for every j there is an i with \[ \sup \{\| U\|^*_ i:\quad u\in F'[U^ 0_ k],\quad \| u\|^*_ j\leq 1\}=\infty \] where \(\|\) \(\|^*_ n\) is the dual norm of the seminorm \(\|\) \(\|_ n\) on F. As one of the simple consequences of the main result the assumption of separability in the theorem stated is removed.
scientific article; zbMATH DE number 4103714