On asymptotically symmetric Banach spaces

Mathematical Proceedings of the Cambridge Philosophical Society, 1993

In , Partington proved that if A is a Banach sequence space with a monotone basis having the Banach-Saks property, and (X n ) is a sequence of Banach spaces each having the Banach-Saks property, then the vector sequence space S A X n has this same property. In addition, Partington gave an example showing that if A and each X n have the weak Banach-Saks property, then S A X n need not have the weak Banach-Saks property.

Journal of the London Mathematical Society, 1991

kuwait journal of science, 2018

Recently, Konca et al. (2015a) have revisited the space p of p -summable sequences of real numbers and have shown that this space is actually contained in a weighted inner product space called 2 v . In another paper, Konca et al. (2015b) have investigated the space p to show that it is also contained in a weighted 2-inner product space. In this work, we show in the details that p is dense in 2 v as a normed space, and as a 2-normed space. Further, we prove that 2 v is separable and conclude that it is isometric to the completion of p .

Journal of Mathematical Analysis and Applications, 2000

Topology and its Applications, 2013

Using a strengthening of the concept of K σδ set, introduced in this paper, we study a certain subclass of the class of K σδ Banach spaces; the so called strongly K σδ Banach spaces. This class of spaces includes subspaces of strongly weakly compactly generated (SWCG) as well as Polish Banach spaces and it is related to strongly weakly Kanalytic (SWKA) Banach spaces as the known classes of K σδ and weakly K-analytic (WKA) Banach spaces are related.

We present an example of a Banach space E admitting an equivalent weakly uniformly rotund norm and such that there is no � : E ! c0(), for any set , linear, one-to-one and bounded. This answers a problem posed by Fabian, Godefroy, Hajek and Zizler. The space E is actually the dual space Yof a space Y which is a subspace of a WCG space.

2013

We prove that a separable Banach space E does not contain a copy of the space c 0 of null-sequences if and only if for every doubly power-bounded operator T on E and for every vector x ∈ E the relative compactness of the sets {T n+m x − T n x : n ∈ N} (for some/all m ∈ N, m ≥ 1) and {T n x : n ∈ N} are equivalent. With the help of the Jacobs-de Leeuw-Glicksberg decomposition of strongly compact semigroups the case of (not necessarily invertible) powerbounded operators is also handled.

1999

We consider Banach space properties that lie between conditions introduced by Bynum and Landes. These properties depend on the metric behavior of weakly convergent sequences. We also investigate the permanence properties of these conditions. Section 3 considers the properties mentioned in the abstract. Section 4 is devoted to permanence results for these properties.

We use a result of Bourgain and Delbaen on extreme points in duals of separable Banach spaces to characterize separable Banach spaces containing isomorphic copies of 1 in terms of extreme points. We also study the weak star closure of a bounded subset A of a separable X Banach space in X * * in terms of the existence of a sequence in A equivalent to the canonical basis of 1 .

2008

Let X be a separable Banach space. We provide an explicit construction of a sequence in X that tends to ∞ in norm but which is weakly dense. Our interest in the result stated in the Abstract was motivated by two theorems. First, in their work on hypercyclic operators, K. Chan and R. Sanders proved the following: Theorem 1. (Chan and Sanders [3]) For any p, 2 ≤ p < ∞, there is a bounded linear operator T : `p(Z) → `p(Z) that is weakly hypercyclic but is not hypercyclic. That is, there is a vector xp ∈ `p(Z) such that {xp, T (xp), · · · , T (xp), · · · } is a weakly dense set and such that for no vector x ∈ `p(Z) is it true that {x, T (x), · · · , T (x), · · · } is norm dense. In fact, the proof in [3] shows the existence of a vector xp such that ‖T (xp)‖ → ∞ while {T (xp) | n ∈ N} is dense in `p(Z) with the weak topology. Moreover, Chan and Sanders remark that, in fact, one can construct a weakly dense sequence (xn) ⊂ `2(Z) such that ‖xn‖ → ∞ ([3, page 49]). Second, V. Kadets has ...