Related papers: A characterization of Banach spaces containing $c_…

A Banach space $X$ has the $2$-summing property if the norm of every linear operator from $X$ to a Hilbert space is equal to the $2$-summing norm of the operator. Up to a point, the theory of spaces which have this property is independent…

Let $\{X, X_{n}; n \geq 1 \}$ be a sequence of i.i.d. $\mathbf{B}$-valued random variables and set $S_{n} = \sum_{i=1}^{n}X_{i},~n \geq 1$. This note is devoted to study the classical central limitr theorem for subsequences of sums of…

We show that any Banach space contains a continuum of non isomorphic subspaces or a minimal subspace. We define an ergodic Banach space $X$ as a space such that $E_0$ Borel reduces to isomorphism on the set of subspaces of $X$, and show…

A basic sequence in a Banach space is called wide-$(s)$ if it is bounded and dominates the summing basis. (Wide-$(s)$ sequences were originally introduced by I.~Singer, who termed them $P^*$-sequences). These sequences and their quantified…

The aim of this paper is to introduce and investigate a new class of separable Banach spaces modeled after an example of Garling from 1968. For each $1\leqslant p<\infty$ and each nonincreasing weight $\textbf{w}\in c_0\setminus\ell_1$ we…

Suppose that $X$ is a Banach space. We will show that $X$ does not contain a copy of $c_0$ if and only if for each series which is not unconditionally convergent in $X$ respective sets coding all bounded subseries and rearrangements are…

A remarkable theorem of R. C. James is the following: suppose that $X$ is a Banach space and $C \subseteq X$ is a norm bounded, closed and convex set such that every linear functional $x^* \in X^*$ attains its supremum on $C$; then $C$ is a…

We consider the class of Banach space $Y$ for which $c_0$ admits a nontrivial twisted sum with $Y$. We present a characterization of such space $Y$ in terms of properties of the $weak^\ast$ topology on $Y^\ast$. We prove that under the…

Let $H$ be a Hilbert space. Using Ball's solution of the "complex plank problem" we prove that the following properties of a sequence $a_n>0$ are equivalent: (1) There is a sequence $x_n \in H$ with $\|x_n\|=a_n$, having 0 as a weak cluster…

Let (e_i) be a dictionary for a separable Banach space X. We consider the problem of approximation by linear combinations of dictionary elements with quantized coefficients drawn usually from a `finite alphabet'. We investigate several…

Let $\lambda$ be a large enough cardinal number (assuming GCH it suffices to let $\lambda=\aleph_\omega$). If $X$ is a Banach space with $\text{dens}(X)\ge\lambda$, which admits a coarse (or uniform) embedding into any $c_0(\Gamma)$, then…

We extend the well-known characterizations of convergence in the spaces $l_p$ ($1\le p<\infty$) of $p$-summable sequence and $c_0$ of vanishing sequences to a general characterization of convergence in a Banach space with a Schauder basis…

This paper is devoted to providing a unifying approach to the study of the uniqueness of unconditional bases, up to equivalence and permutation, of infinite direct sums of quasi-Banach spaces. Our new approach to this type of problem…

We first include a result of the second author showing that the Banach space S is complementably minimal. We then show that every block sequence of the unit vector basis of S has a subsequence which spans a space isomorphic to its square.…

Let X be a Banach space. Given a subset A of the dual space X* denote by $A_{(1)}$ the weak* sequential closure of A, i.e., the set of all limits of weak*-convergent sequences in A. The study of weak* sequential closures of linear subspaces…

A Banach space $\X$ has the complete continuity property (CCP) if each bounded linear operator from $L_1$ into $\X$ is completely continuous (i.e., maps weakly convergent sequences to norm convergent sequences). The main theorem shows that…

Our aim in this article is to contribute to the study of the structure of subsymmetric basic sequences in Banach spaces (even, more generally, in quasi-Banach spaces). For that we introduce the notion of positioning and develop new tools…

The class of countably intersected families of sets is defined. For any such family we define a Banach space not containing $\ell^{1}(\NN )$. Thus we obtain counterexamples to certain questions related to the heredity problem for W.C.G.…

We characterize noncommutative symmetric Banach spaces for which every bounded sequence admits either a convergent subsequence, or a $2$-co-lacunary subsequence. This extends the classical characterization, due to R\"abiger.

We prove that given a locally compact Hausdorff space, $K$, and a compact C$^*$-algebra, $\mathcal{A}$, the C$^*$-algebra $C(K, \mathcal{A})$ satisfies that every convex combination of slices of the closed unit ball is relatively weakly…