Quantification of Banach-Saks properties of higher orders (Q2674327)

Recall that a Banach space is said to have the \textit{(weak) Banach-Saks property} if every bounded (weakly null) sequence has a Cesàro-summable subsequence, that is, a subsequence with norm-converging averages. The difference between the weak Banach-Saks property and the Mazur theorem, which gives the convergence of a convex combination (not neccessarily arithmetic means as in the weak Banach-Saks property), motivated the authors of [\textit{S.~Argyros} et al., Isr. J. Math. 107, 157--193 (1998; Zbl 0942.46007)] to define the (weak) Banach-Saks property of higher orders. The author deals with a quantified version of a variant of the \(\xi\)-Banach-Saks property and weak \(\xi\)-Banach-Saks property for countable ordinals \(\xi\). He suggests several ways of quantitative versions of those properties and very deeply investigates those. Quantification of the (weak) Banach-Saks property has already been handled in [\textit{H.~Bendová} et. al., J. Funct. Anal. 268, 1733--1754 (2015; Zbl 1357.46009)], the generalization to the (weak) Banach-Saks property of higher order is nontrivial and much more technically and combinatorially involved. The main results include improvements of some deep results proved by Argyros et al. [loc.\,cit.], or a negative answer to a question published in [Bendová et al., loc.\,cit.]. The author also suggests new interesting questions/problems suitable for further investigations. Reviewer's remark. Let me note that, as confirmed by the author, there is a gap in the proof of Lemma 5.3. Namely, on p.\,21, line 24 we have \(\zeta_k^P = [\zeta_{n_k}]_1^{P_k}\) where \(n_k=\min P_k\) (and not \([\zeta_{k}]_1^{P_k}\) as the author claims) -- this affects the remainder of the proof and at the moment it is not clear how to fix it. Therefore, it is not clear at the moment whether Lemma 5.3 holds and consequently, whether Proposition 5.4 holds. The remainder of the paper is not affected by this gap (the only difference is that instead of \(\min\) we should write \(\inf\) in the definition of \(\delta_0(A)\) on p.\,24).