The Littlewood-Orlicz operator ideal
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 \textit{J.\,E.\thinspace Littlewood} in [Q.\ J.\ Math., Oxf.\ Ser.\ 1, 164--174 (1930; JFM 56.0335.01)] and \textit{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 \textit{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.
injective and projective tensor products
scientific article; zbMATH DE number 5652725
scientific article