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2-quasi-\(\lambda\)-nuclear maps. (English) Zbl 1096.47025

It is a well-known result due to A. Pietsch that the composition of two quasi-nuclear mappings between Banach spaces is nuclear. Using the same methods, the author proves as the main result of this paper that the product of two quasi-\(\lambda\)-nuclear mappings is pseudo-\(\lambda\)-nuclear for all sequence spaces \(\lambda \subseteq \ell^1\). Here, a map \(T:E\to F\) is said to be pseudo-\(\lambda\)-nuclear if it admits a representation \(Tx=\sum\limits_{n=1}^\infty \alpha_n\langle x, a_n \rangle y_n\) for some sequence \(\alpha=(\alpha_n) \in \lambda\) and some bounded sequences \((a_n)\) and \((y_n)\). The map \(T\) is called quasi-\(\lambda\)-nuclear if it allows an estimation \(\| Tx\| \leq \sum\limits_{n=1}^\infty | \alpha_n| \cdot | \langle x, a_n \rangle| \). The key point in the proof is the fact that quasi-\(\lambda\)-nuclear maps with values in \(\ell^\infty(I)\) are automatically pseudo-\(\lambda\)-nuclear.

MSC:

47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
46A45 Sequence spaces (including Köthe sequence spaces)
47L20 Operator ideals
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