Banach–Saks property in Marcinkiewicz spaces

Abstract

We catalogue all Marcinkiewicz function and sequence spaces with the Banach-Saks property and present necessary and sufficient conditions for a wide subclass of spaces to possess the p-Banach-Saks property, 1 < p < ∞. We apply our results to several open problems.

Structure of Cesàro function spaces: a survey

“…Therefore, if I = [0, 1], the second equality in (14) can be proved in the same way as in the part (a) by using reiteration formulae (11) and (12).…”

“…Note that the proof of the second equality in (14) for the spaces on I = [0, ∞) in [11] is essentially based on some results from the paper [71]. Moreover, another proof of this equality in the case I = [0, ∞) was also given by Sinnamon [ …”

The Banach-Saks index of some sequence spaces

“…Moreover, the l 0 p,∞ spaces, 1 < p < ∞, satisfy γ l 0 p,∞ = p [12] and γ(l p ,1 ) = γ l 0 p,∞ * = p [13]; i.e.,…”

Cesaro mean convergence of martingale differences in rearrangement invariant spaces

“…To prove that (iii) =⇒ (i), assume that (i) does not hold. Thanks to [4], we then have that the separable part M 0 (ϕ) of the Marcinkiewicz space M (ϕ) does not have the Banach-Saks property. This immediately implies that one can locate in the space M 0 (ϕ) a weakly null sequence of martingale differences whose Cesaro means do not converge strongly.…”

“…Later, it was shown by Szlenk [33] that the non-uniformly convex Banach space L 1 [0, 1) also has the Banach-Saks property. More recently, Banach-Saks property and Banach-Saks type properties have been actively studied in the class of separable rearrangement invariant (r.i.) function spaces X with the Fatou property [10], [9], [32], [31], [3], [4] (all relevant definitions are given below). A general approach employed in these articles goes back to the paper [15] (see also the paper of Gaposhkin [13]) which studied weakly null sequences and martingale differences in classical L p -spaces.…”