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

Bu, Qingying; Ji, Donghai; Wang, Yuwen

Rocky Mt. J. Math. 39, No. 6, 1847-1857 (2009).

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.

Reviewer: Carsten Michels (Oldenburg)

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

Keywords:

Littlewood-Orlicz operators; weakly compact operators; injective and projective tensor products; \(\mathcal{L}_p\)-spaces; absolutely summing operators

Citations:

Zbl 1041.46006; Zbl 1112.47014; Zbl 0008.31501; Zbl 0434.47030; JFM 56.0335.01; JFM 59.1076.03

References:

[1]	H. Apiola, Duality between spaces of \(p\)-summable sequences, \((p,q)\)-summing operators and characterization of nuclearity , Math. Ann. 219 (1976), 53-64. · Zbl 0304.47025
[2]	Q. Bu, On Banach spaces verifying Grothendieck’s theorem , Bull. London Math. Soc. 35 (2003), 738-748. · Zbl 1041.46006
[3]	Q. Bu and J. Diestel, Observations about the projective tensor product of Banach spaces , I \(\A\), \(1 < p < \infty\), Quaest. Math. 24 (2001), 519-533. · Zbl 1038.46061
[4]	J.S. Cohen, Absolutely \(p\)-summing, \(p\)-nuclear operators, and their conjugates , Math. Ann. 201 (1973), 177-200. · Zbl 0233.47019
[5]	A. Defant and K. Floret, Tensor norms and operator ideals , North-Holland, Amsterdam, 1993. · Zbl 0774.46018
[6]	J. Diestel, Sequences and series in Banach spaces , Grad. Texts Math. 92 , Springer-Verlag, New York, 1984. · Zbl 0542.46007
[7]	J. Diestel, J. Fourie and J. Swart, A theorem of Littlewood, Orlicz, and Grothendieck about sums in \(L^1(0,1)\) , J. Math. Anal. Appl. 251 (2000), 376-394. · Zbl 0974.46031
[8]	J. Diestel, H. Jarchow and A. Tonge, Absolutely summing operators , Cambridge University Press, Cambridge, 1995. · Zbl 0855.47016
[9]	A. Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques , Bol. Soc. Mat. São Paulo 8 (1953/1956), 1-79. · Zbl 0074.32303
[10]	——–, Sur certaines classes des suites dans les espaces de Banach, et le théorème de Dvoretzky-Rogers , Bol. Soc. Mat. São Paulo 8 (1953/1956), 81-110.
[11]	S. Kwapien, Some remarks on \((p,q)\)-summing operators in \(\ell_p\)-spaces , Stud. Math. 29 (1968), 327-337. · Zbl 0182.17001
[12]	J.E. Littlewood, On bounded bilinear forms in an infinite number of variables , Quart. J. Math. (Oxford) 1 (1930), 164-174. · JFM 56.0335.01
[13]	B.S. Mityagin and A. Pelczynski, Nuclear operators and approximative dimensions , Proc. Inter. Congress of Mathematicians, Moscow, 1966. · Zbl 0191.41704
[14]	W. Orlicz, it Über unbedingte Konvergenz in Funktionenräumem (I), Stud. Math., 4 (1933), 33-37. · Zbl 0008.31501
[15]	A. Pelczynski, Banach spaces on which every unconditionally converging operator is weakly compact , Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 10 (1962), 641-648. · Zbl 0107.32504
[16]	A. Pietsch, Absolut \(p\)-summierende abbildungen in normierten räumen , Stud. Math. 28 (1967), 333-353. · Zbl 0156.37903
[17]	——–, Operator ideals , North-Holland, Amsterdam, 1980.