Rearrangement invariant spaces satisfying Dunford-Pettis criterion of weak compactness. (English) Zbl 1443.46018
Kuchment, Peter (ed.) et al., Functional analysis and geometry. Selim Grigorievich Krein centennial. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 733, 45-59 (2019).
Summary: We survey old and recent results related to rearrangement invariant spaces \(X\), where an analogue of the classical Dunford-Pettis characterization of relatively weakly compact subsets in \(L_1\) holds. This is precisely the class of spaces such that for every weakly null sequence \(\{x_n\}_{n=1}^\infty \subset X\) of pairwise disjoint functions we have \(\|x_n\|_X\to 0\). In these spaces, Cesàro means of any weakly null martingale difference sequence are norm null. Moreover, all reflexive subspaces of such a space \(X\) are strongly embedded in \(X\). We simplify the proofs of some known results and prove a few new ones.
For the entire collection see [Zbl 1420.35010].
For the entire collection see [Zbl 1420.35010].
MSC:
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 46B50 | Compactness in Banach (or normed) spaces |
| 46B42 | Banach lattices |
Keywords:
weakly compact set; equi-integrable set; equi-absolutely continuous norms; rearrangement invariant space; Orlicz space; Marcinkiewicz space; Lorentz space; positive Schur property; martingale difference; strongly embedded subspace; reflexive subspaceReferences:
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