\(\lambda\)-nuclear spaces which are nuclear (Q749817)

Let \(\lambda\) be a normal sequence space. A linear map T: \(E\to F\) between Banach spaces is called \(\lambda\)-nuclear if it can be written in the form \(Tx=\sum^{\infty}_{n=1}\xi_ nu_ n(x)y_ n\) where \((\xi_ n)\in \lambda\), \((u_ n)\) is bounded sequence in \(E'\) and \((v(y_ n))\in \lambda^{\times}\) for every \(v\in F'\). It is proved that if E and F are Hilbert spaces, then T is \(\ell_{\infty}\)-nuclear if and only if it is a Hilbert-Schmidt map. Hence if a locally convex space E is \(\lambda\)-nuclear in the sense of \textit{E. Dubinsky} and \textit{M. S. Ramanujan} [Mem. Am. Math. Soc. 128 (1971; Zbl 0244.47015)] then E is nuclear if and only if E is Hilbertian.