Related papers: Cotype and summing properties in Banach spaces

A recent result of T.~Abrahamsen, P.~H\'ajek and S.~Troyanski states that a separable Banach space is almost square if and only if there exists $h\in S_{X^{****}}$ such that $\|x+h\|=\max\{\|x\|,1\}$ for all $x\in X$. The proof passes…

We study the problem of the existence of a common algebraic complement for a pair of closed subspaces of a Banach space. We prove the following two characterizations: (1) The pairs of subspaces of a Banach space with a common complement…

Unique continuation properties for a class of evolution equations defined on Banach spaces are considered from two different point of views: the first one is based on the existence of conserved quantities, which very often translates into…

Let $E$ be a Frechet-Montel space and $(E_n)_{n \in \mathbb{N}}$ be a finite dimensional unconditional decomposition of $E$ with $\dim(E_n)\leq k$ for some fixed $k \in \mathbb{N}$ and for all $n \in \mathbb{N}$. Consider a sequence…

A well-known result of R. Pol states that a Banach space $X$ has property ($\mathcal{C}$) of Corson if and only if every point in the weak*-closure of any convex set $C \subseteq B_{X^*}$ is actually in the weak*-closure of a countable…

A necessary and sufficient condition for existence of a Banach space with a finite dimensional decomposition but without the $\pi$-property in terms of norms of compositions of projections is found.

Combining the methods of Brian and Stuart with the classical Dvoretzky theorem, we show that no infinite-dimensional Banach space contains a barrelled subspace of (algebraic) dimension $<\mbox{cov}(\mathcal{N})$, the covering number of the…

A Banach space $X$ with a Schauder basis is defined to have the restricted quotient hereditarily indecomposable (QHI) property if $X/Y$ is hereditarily indecomposable (HI) for any infinite codimensional subspace $Y$ with a successive…

Following results of Bourgain and Gorelik we show that the spaces $\ell_p$, $1<p<\infty$, as well as some related spaces have the following uniqueness property: If $X$ is a Banach space uniformly homeomorphic to one of these spaces then it…

Suppose that $E$ and $E'$ denote real Banach spaces with dimension at least 2, that $D\subset E$ and $D'\subset E'$ are domains, and that $f: D\to D'$ is a homeomorphism. In this paper, we prove the following subinvariance property for the…

Given a Banach space $X$ and a real number $\alpha\ge 1$, we write: (1) $D(X)\le\alpha$ if, for any locally finite metric space $A$, all finite subsets of which admit bilipschitz embeddings into $X$ with distortions $\le C$, the space $A$…

Basis of a Banach space with respect to a filter F on N (F-basis for short) is a generalization of basis, where the ordinary convergence of series is substituted by convergence of partial sums with respect to the filter F. We study the…

We develop a novel and unifying setting for phase retrieval problems that works in Banach spaces and for continuous frames and consider the questions of uniqueness and stability of the reconstruction from phaseless measurements. Our main…

Naor and Mendel's metric cotype extends the notion of the Rademacher cotype of a Banach space to all metric spaces. Every Banach space has metric cotype at least 2. We show that any metric space that is bi-Lipschitz equivalent to an…

We show that, if $X$ is a closed subspace of a Banach space $E$ and $Z$ is a closed subspace of $E^*$ such that $Z$ is norming for $X$ and $X$ is total over $Z$ (as well as $X$ is norming for $Z$ and $Z$ is total over $X$), then $X$ and the…

A Banach space $X$ is said to have the ball generated property (BGP) if every closed, bounded, convex subset of $X$ can be written as an intersection of finite unions of closed balls. In 2002 S. Basu proved that the BGP is stable under…

The article studies the convergence of trigonometric Fourier series via a new Tauberian theorem for Ces\`{a}ro summable series in abstract normed spaces. This theorem generalizes some known results of Hardy and Littlewood for number series.…

This paper contains two improvements on a theorem of S. N. Bernstein for Banach spaces. We show that if $X$ is an arbitrary infinite-dimensional Banach space, $\{Y_n\}$ is a sequence of strictly nested subspaces of $ X$ and if $\{d_n\}$ is…

Given a $1$-cocycle $b$ with coefficients in an orthogonal representation, we show that any finite dimensional summand of $b$ is cohomologically trivial if and only if $\| b(X_n) \|^2/n$ tends to a constant in probability, where $X_n$ is…

We define and study asymptotically symmetric Banach spaces (a.s.) and its variations: weakly a.s. (w.a.s.) and weakly normalized a.s. (w.n.a.s.). If X is a.s. then all spreading models of X are uniformly symmetric. We show that the converse…