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