On the structure of the set of higher order spreading models (Q2921801)

Spreading models were established by \textit{A. Brunel} and \textit{L. Sucheston} [Math. Syst. Theory 7(1973), 294--299 (1974; Zbl 0323.46018)] and since their appearance they play a prominent role in the asymptotic theory of Banach spaces. Recently, \textit{S. A. Argyros} et al. [Fundam. Math. 221, No. 1, 23--68 (2013; Zbl 1296.46010)] provided a wide extension of the concept of a spreading model. In particular, given any Banach space \(X\) and any countable ordinal \(\xi\), they introduced the spreading models of order \(\xi\) associated to the space \(X\).NEWLINENEWLINEThe definition of the higher-order spreading models is based on the notion of \(\mathcal{F}\)-sequences and on plegma families. An \(\mathcal{F}\)-sequence in the space \(X\) is a sequence of the form \((x_s)_{s\in\mathcal{F}}\), where \(x_s\in X\) and the index set \(\mathcal{F}\) is a regular thin family of finite subsets of \(\mathbb{N}\). (A family of finite subsets of \(\mathbb{N}\) is called regular thin if it satisfies some specific properties. Any regular thin family \(\mathcal{F}\) is associated with a countable ordinal \(o(\mathcal{F})\) which measures the complexity of \(\mathcal{F}\) and is called the order of \(\mathcal{F}\).) A plegma family is a sequence \((s_1,s_2,\dots,s_l)\) of nonempty finite subsets of \(\mathbb{N}\) where the first elements of \(s_1,s_2,\dots,s_l\) are in increasing order and they lie before their second elements which are also in increasing order and so on. Now, given a sequence \((e_n)\) in a seminormed space \((E,\|\cdot\|_\ast)\), we say that \((e_n)\) is a spreading model of order \(\xi\) of the Banach space \(X\) if there are a regular thin family \(\mathcal{F}\) of order \(\xi\), an \(\mathcal{F}\)-sequence \((x_s)_{s\in\mathcal{F}}\) in \(X\) and an infinite subset \(M\) of \(\mathbb{N}\) such that for some null sequence \((\delta_n)_{n\in\mathbb{N}}\) of positive real numbers we have \(\left|\big\|\sum_{i=1}^k a_i x_{s_i}\big\|- \big\|\sum_{i=1}^k a_i e_i\big\|_\ast \right| \leq \delta_l\) for any \(l\in \mathbb{N}\), any \(1\leq k\leq l\), any \(a_1,a_2,\dots,a_n\in [-1,1]\) and any \((s_1,s_2,\dots,s_k)\) which is a plegma family with \(s_i\in\mathcal{F}\), \(s_i\subset M\) and \(\min s_1 \geq M(l)\).NEWLINENEWLINELet \(SM_\xi(X)\) denote the set of all spreading models of order \(\xi\) associated to the space \(X\). Then \(SM_1(X)\) coincides with the classical spreading models. Argyros et al. [loc. cit.] proved that \((SM_\xi(X))_{\xi<\omega_1}\) is an increasing transfinite hierarchy and they studied the fundamental properties of this hierarchy.NEWLINENEWLINEThe purpose of the paper under review is to shed light on the structure of the subset \(SM_\xi^w(X)\) of \(SM_\xi(X)\) which contains all \(\xi\)-order spreading models generated by subordinated weakly null \(\mathcal{F}\)-sequences. Roughly speaking, subordinated \(\mathcal{F}\)-sequences are a higher-order analogue of ordinary weakly convergent sequences. The set \(SM_\xi^w(X)\) is endowed with the pre-partial order of domination. That is, if \((e_n^1), (e_n^2) \in SM_\xi^w(X)\), then \((e_n^1)\leq (e_n^2)\) if there is a constant \(C>0\) such that \(\|\sum_{j=1}^n a_j e_j^1\| \leq C \|\sum_{j=1}^n a_j e_j^2\|\) for any \(n\in\mathbb{N}\) and any scalars \(a_1,a_2,\dots,a_n\). This binary relation is not a partial order. However, if we identify equivalent \(\xi\)-order spreading models, then the set \(\mathcal{SM}_\xi^w(X)\) of all equivalent classes becomes a partial order set.NEWLINENEWLINEThe authors show that \((SM_\xi^w(X))_{\xi<\omega_1}\) forms an increasing transfinite hierarchy. In the main part of the paper, they prove that several interesting results concerning classical spreading models generated by weakly null sequences can be extended to the higher-order setting. The main results of the paper are described below.NEWLINENEWLINEFirstly, the authors prove that for any countable ordinal \(\xi\) the set \(\mathcal{SM}_\xi^w(X)\) is an upper semilattice. Furthermore, it is shown that every countable subset of \(SM_\xi^w(X)\) admits an upper bound in \(SM_\xi^w(X)\). (The corresponding result for classical spreading models generated by weakly null sequences is contained in [\textit{G. Androulakis} et al., Can. J. Math. 57, No. 4, 673--707 (2005; Zbl 1090.46004)].)NEWLINENEWLINESecondly, it is proved that if \(SM_\xi^w(X)\) contains a strictly increasing sequence of length \(\omega\), then \(SM_\xi^w(X)\) contains a strictly increasing sequence of length \(\omega_1\). (The corresponding result for classical spreading models is proved in [\textit{B. Sari}, Proc. Am. Math. Soc. 134, No. 5, 1339--1345 (2006; Zbl 1098.46012)].)NEWLINENEWLINEThirdly, under the additional assumption that the space \(X\) has separable dual, the authors prove the following results concerning the set \(SM_\xi^w(X)\): \((a)\) If \(\mathcal{SM}_\xi^w(X)\) is uncountable, then there exists a subset of \(SM_\xi^w(X)\) of size continuum such that the elements of this set are pairwise incomparable. \((b)\) If \(SM_\xi^w(X)\) contains a strictly decreasing sequence of length \(\omega_1\), then it contains a strictly increasing sequence of length \(\omega_1\). \((c)\) If \(SM_\xi^w(X)\) does not contain a strictly increasing sequence of length \(\omega\), then there exists a countable ordinal \(\zeta\) such that \(SM_\xi^w(X)\) does not contain any decreasing sequence of length \(\zeta\). (The corresponding results for classical spreading models hold under the weaker assumption that \(X\) is separable [\textit{P. Dodos}, Can. Math. Bull. 53, No. 1, 64--76 (2010; Zbl 1197.46009)].)NEWLINENEWLINEFinally, the authors pose the question if the aforementioned results \((a),(b)\) and \((c)\) hold without the separable dual assumption. Towards this direction, they prove that these results remain valid in the case where \(X\) is separable and admits no spreading model of order one equivalent to the standard basis of \(\ell_1\).