Unconditional structures of weakly null sequences

The following dichotomy is established for a normalized weakly null sequence in a Banach space: Either every subsequence admits a convex block subsequence equivalent to the unit vector basis of , or there exists a subsequence which is boundedly convexly complete.

     

Positional graphs and conditional structure of weakly null sequences

We prove that, unless assuming additional set theoretical axioms, there are no reflexive spaces without unconditional sequences of the density continuum. We show that for every integer n$n$ there are normalized weakly-null sequences of length _ωn${\omega }_n$ without unconditional subsequences. This together with a result of Dodos et al. (2011) [7] shows that _ωω${\omega }_{\omega }$ is the minimal cardinal κ$\kappa$ that could possibly have the property that every weakly null κ$\kappa$-sequence has an infinite unconditional basic subsequence. We also prove that for every cardinal number κ$\kappa$ which is smaller than the first ω$\omega$-Erd?s cardinal there is a normalized weakly-null sequence without subsymmetric subsequences. Finally, we prove that mixed Tsirelson spaces of uncountable densities must always contain isomorphic copies of either _c0$c_0$ or _?p${?}_p$, with p≥1$p\ge 1$.     

The central limit theorem for summability methods of some weakly dependent sequences

We consider the asymptotic normality of the random variable Z(λ) = Σc_j(λ)ζ_jasλ → ∞ where {ζ_j} is a strictly stationary strongly mixing sequence.     

Pre-compact families of finite sets of integers and weakly null sequences in Banach spaces

In this paper we use the Nash-Williams theory of fronts and barriers to study weakly null sequences in Banach spaces. Specifically, we show how barriers relate to the classical fact that C(K) with K a countable compactum is c₀-saturated. Another result relates the notion of a barrier to the Maurey-Rosenthal example of a weakly null sequence with no unconditional subsequences. In particular, we construct examples of weakly-null sequences which are α-unconditional but not β-unconditional.     

A summability factor theorem for absolute summability involving quasi-monotone sequences

In this paper we prove a theorem on _|A|k,k≥1${\left|A\right|}_k,k\ge 1$, summability factors for an infinite series by replacing a weighted mean matrix with a triangular matrix.     

Two valued measure and summability of double sequences

In this paper, following the methods of Connor [2], we extend the idea of statistical convergence of a double sequence (studied by Muresaleen and Edely [12]) to μ-statistical convergence and convergence in μ-density using a two valued measure μ. We also apply the same methods to extend the ideas of divergence and Cauchy criteria for double sequences. We then introduce a property of the measure μ called the (APO₂) condition, inspired by the (APO) condition of Connor [3]. We mainly investigate the interrelationships between the two types of convergence, divergence and Cauchy criteria and ultimately show that they become equivalent if and only if the measure μ has the condition (APO₂).     

Exhaustive weakly wandering sequences

A sequence of integers {n_i : i = 0, 1…} is an exhaustive weakly wandering sequence for a transformation T if for some measurable set W, X=^∞_i=0TⁿⁱW(disj. We introduce a hereditary Property (H) for a sequence of integers associated with an infinite ergodic transformation T, and show that it is a sufficient condition for the sequence to be an exhaustive weakly wandering sequence for T. We then show that every infinite ergodic transformation admits sequences that possess Property (H), and observe that Property (H) is inherited by all subsequences of a sequence that possess it. As a corollary, we obtain an application to tiling the set of integers with infinite subsets.     

Bornological spaces of null sequences

The research of the second author was partially supported by the Committee of Scientific Research (KBN), Poland, grant 2 P301 003 07.     

On weakly null FDD'S in Banach spaces

In this paper we show that every sequence (F _n ) of finite dimensional subspaces of a real or complex Banach space with increasing dimensions can be “refined” to yield an F.D.D. (G _n ), still having increasing dimensions, so that either every bounded sequence (x _n ), withx _n ∈G _n forn∈ℕ, is weakly null, or every normalized sequence (x _n ),withx _n ∈G _n forn∈ℕ, is equivalent to the unit vector basis of ℓ₁. Crucial to the proof are two stabilization results concerning Lipschitz functions on finite dimensional normed spaces. These results also lead to other applications. We show, for example, that every infinite dimensional Banach spaceX contains an F.D.D. (F _n ),with lim _n→∞dim(F _n )=∞, so that all normalized sequences (x _n ),withx _n ∈F _n ,n∈∕, have the same spreading model overX. This spreading model must necessarily be 1-unconditional overX. Research partially supported by NSF DMS-8903197, DMS-9208482, and TARP 235. Research partially supported by NSF.     

General absolute summability factor theorems involving quasi-power-increasing sequences

In this paper a general theorem concerning the |A,δ|_k summability methods has been proved, which generalizes two results of Şevli and Leindler [H. Şevli and L. Leindler, On the absolute summability factors of infinite series involving quasi-power-increasing sequences, Computers and Mathematics with Applications 57 (2009), 702–709]. We obtain sufficient conditions for ∑a_nλ_n to be summable |A,δ|_k, k1, 0δ<1/k, by using quasi-power-increasing sequences.     

Two valued measure and summability of double sequences in asymmetric context

We extend the ideas of convergence and Cauchy condition of double sequences extended by a two valued measure (called ??-statistical convergence/Cauchy condition and convergence/Cauchy condition in ??-density, studied for real numbers in our recent paper [7]) to a very general structure like an asymmetric (quasi) metric space. In this context it should be noted that the above convergence ideas naturally extend the idea of statistical convergence of double sequences studied by Móricz [15] and Mursaleen and Edely [17]. We also apply the same methods to introduce, for the first time, certain ideas of divergence of double sequences in these abstract spaces. The asymmetry (or rather, absence of symmetry) of asymmetric metric spaces not only makes the whole treatment different from the real case [7] but at the same time, like [3], shows that symmetry is not essential for any result of [7] and in certain cases to get the results, we can replace symmetry by a genuinely asymmetric condition called (AMA).     

Indecomposability and unconditionality in duality

We construct a reflexive and unconditionally saturated Banach space such that its dual is Hereditarily Indecomposable. We also show that every quotient of has a further quotient which is Hereditarily Indecomposable and every quotient of has a further quotient with an unconditional basis.     

The set of almost convergent sequences as intersections of summability fields

Archiv der Mathematik -     

Elton near unconditionality of arrays

There are many notions of partial unconditionality defined for a weakly null basic sequence in a Banach space. In 2008 the idea of Schreier unconditionality was extended to a structure in Banach spaces called arrays. Here we extend the idea of Elton near unconditionality to arrays.     

Weakly null sequences with an unconditional subsequence

In the present paper we provide sufficient conditions such that a normalized pointwise convergent to zero sequence in with a compact space and a Banach space has an unconditional subsequence.

As a consequence we obtain that any such sequence of functions with finite and uniformly bounded cardinality of their range admits an unconditional subsequence.