## The Littlewood-Orlicz operator ideal.(English)Zbl 1186.47018

A continuous linear operator $$u:X\to Y$$ between Banach spaces $$X$$ and $$Y$$ is called a Littlewood-Orlicz operator if $$\text{id}\otimes u$$ maps (continuously) the injective tensor product of $$\ell_1$$ with $$X$$ into the projective tensor of $$\ell_2$$ with $$Y$$. The naming of this class of operators is motivated by two classical results due to J. E. Littlewood in [Q. J. Math., Oxf. Ser. 1, 164–174 (1930; JFM 56.0335.01)] and W. Orlicz in [Stud. Math. 4, 33–37 (1933; Zbl 0008.31501, JFM 59.1076.03)], respectively, that the identity operators on $$\ell_1$$ and $$L_1[0,1]$$, respectively, have this property. The notion of Littlewood-Orlicz operators coincides with the one of $$(1,1,2)$$-summing operators introduced by A. Pietsch in [“Operator ideals” (North-Holland Mathematical Library 20) (1980; Zbl 0434.47030)].
The first named author of the article under review used in [Bull. Lond. Math. Soc. 35, No. 6, 738–748 (2003; Zbl 1041.46006)] Littlewood-Orlicz operators to characterize Grothendieck type spaces of cotype $$2$$. In the paper under review, the authors prove some properties of Littlewood-Orlicz operators between $$\mathcal{L}_p$$-spaces as well as a result on the weak compactness of Littlewood-Orlicz operators starting from a $$C(K)$$-space.
It might be worth mentioning that from an article by the reviewer [Stud. Math. 178, No. 1, 19–45 (2007; Zbl 1112.47014)] on absolutely $$(r,p,q)$$-summing inclusion maps, one may extract further results on Littlewood-Orlicz operators: a characterization of when the inclusion map $$\text{id}: \ell_u \rightarrow \ell_v$$ is a Littlewood-Orlicz operator; the limit order of the associated Banach operator ideal; the asymptotic decay of the sequence of Hilbert numbers of a Littlewood-Orlicz operator.

### MSC:

 47B10 Linear operators belonging to operator ideals (nuclear, $$p$$-summing, in the Schatten-von Neumann classes, etc.) 46B28 Spaces of operators; tensor products; approximation properties
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### References:

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