Weakly null sequences with upper estimates

Daniel Freeman

Displaying similar documents to “Weakly null sequences with upper estimates”

$L$ -limited-like properties on Banach spaces

Ioana Ghenciu (2023)

Commentationes Mathematicae Universitatis Carolinae

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We study weakly precompact sets and operators. We show that an operator is weakly precompact if and only if its adjoint is pseudo weakly compact. We study Banach spaces with the $p$ - $L$ -limited $^{*}$ and the $p$ -(SR $^{*}$ ) properties and characterize these classes of Banach spaces in terms of $p$ - $L$ -limited $^{*}$ and $p$ -Right $^{*}$ subsets. The $p$ - $L$ -limited $^{*}$ property is studied in some spaces of operators.

Property $(𝐰𝐋)$ and the reciprocal Dunford-Pettis property in projective tensor products

Ioana Ghenciu (2015)

Commentationes Mathematicae Universitatis Carolinae

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A Banach space $X$ has the reciprocal Dunford-Pettis property ( $R D P P$ ) if every completely continuous operator $T$ from $X$ to any Banach space $Y$ is weakly compact. A Banach space $X$ has the $R D P P$ (resp. property $(w L)$ ) if every $L$ -subset of $X^{*}$ is relatively weakly compact (resp. weakly precompact). We prove that the projective tensor product $X \otimes_{π} Y$ has property $(w L)$ when $X$ has the $R D P P$ , $Y$ has property $(w L)$ , and $L (X, Y^{*}) = K (X, Y^{*})$ .

On asymptotically symmetric Banach spaces

M. Junge, D. Kutzarova, E. Odell (2006)

Studia Mathematica

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A Banach space X is asymptotically symmetric (a.s.) if for some C < ∞, for all m ∈ ℕ, for all bounded sequences ${(x_{j}^{i})}_{j = 1}^{\infty} \subseteq X$ , 1 ≤ i ≤ m, for all permutations σ of 1,...,m and all ultrafilters ₁,...,ₘ on ℕ, $l i m_{n ₁, ₁} . . . l i m_{n ₘ, ₘ} | | \sum_{i = 1}^{m} x_{n_{i}}^{i} | | \leq C l i m_{n_{σ (1)},_{σ (1)}} . . . l i m_{n_{σ (m)},_{σ (m)}} | | \sum_{i = 1}^{m} x_{n_{i}}^{i} | |$ . We investigate a.s. Banach spaces and several natural variations. X is weakly a.s. (w.a.s.) if the defining condition holds when restricted to weakly convergent sequences ${(x_{j}^{i})}_{j = 1}^{\infty}$ . Moreover, X is w.n.a.s. if we restrict the condition further to normalized weakly null sequences. If X is a.s. then...

Weak precompactness and property (V*) in spaces of compact operators

Ioana Ghenciu (2015)

Colloquium Mathematicae

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We give sufficient conditions for subsets of compact operators to be weakly precompact. Let $L_{w *} (E *, F)$ (resp. $K_{w *} (E *, F)$ ) denote the set of all w* - w continuous (resp. w* - w continuous compact) operators from E* to F. We prove that if H is a subset of $K_{w *} (E *, F)$ such that H(x*) is relatively weakly compact for each x* ∈ E* and H*(y*) is weakly precompact for each y* ∈ F*, then H is weakly precompact. We also prove the following results: If E has property (wV*) and F has property (V*), then $K_{w *} (E *, F)$ has property (wV*). Suppose...

Limited $p$ -converging operators and relation with some geometric properties of Banach spaces

Mohammad B. Dehghani, Seyed M. Moshtaghioun (2021)

Commentationes Mathematicae Universitatis Carolinae

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By using the concepts of limited $p$ -converging operators between two Banach spaces $X$ and $Y$ , $L_{p}$ -sets and $L_{p}$ -limited sets in Banach spaces, we obtain some characterizations of these concepts relative to some well-known geometric properties of Banach spaces, such as $*$ -Dunford–Pettis property of order $p$ and Pelczyński’s property of order $p$ , $1 \leq p < \infty$ .

A note on Dunford-Pettis like properties and complemented spaces of operators

Ioana Ghenciu (2018)

Commentationes Mathematicae Universitatis Carolinae

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Equivalent formulations of the Dunford-Pettis property of order $p$ ( ${D P P}_{p}$ ), $1 < p < \infty$ , are studied. Let $L (X, Y)$ , $W (X, Y)$ , $K (X, Y)$ , $U (X, Y)$ , and $C_{p} (X, Y)$ denote respectively the sets of all bounded linear, weakly compact, compact, unconditionally converging, and $p$ -convergent operators from $X$ to $Y$ . Classical results of Kalton are used to study the complementability of the spaces $W (X, Y)$ and $K (X, Y)$ in the space $C_{p} (X, Y)$ , and of $C_{p} (X, Y)$ in $U (X, Y)$ and $L (X, Y)$ .

On Hattori spaces

A. Bouziad, E. Sukhacheva (2017)

Commentationes Mathematicae Universitatis Carolinae

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For a subset $A$ of the real line $ℝ$ , Hattori space $H (A)$ is a topological space whose underlying point set is the reals $ℝ$ and whose topology is defined as follows: points from $A$ are given the usual Euclidean neighborhoods while remaining points are given the neighborhoods of the Sorgenfrey line. In this paper, among other things, we give conditions on $A$ which are sufficient and necessary for $H (A)$ to be respectively almost Čech-complete, Čech-complete, quasicomplete, Čech-analytic and weakly separated...

Application of $(L)$ sets to some classes of operators

Kamal El Fahri, Nabil Machrafi, Jawad H&#039;michane, Aziz Elbour (2016)

Mathematica Bohemica

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The paper contains some applications of the notion of $Ł$ sets to several classes of operators on Banach lattices. In particular, we introduce and study the class of order $(L)$ -Dunford-Pettis operators, that is, operators from a Banach space into a Banach lattice whose adjoint maps order bounded subsets to an $(L)$ sets. As a sequence characterization of such operators, we see that an operator $T : X \to E$ from a Banach space into a Banach lattice is order $Ł$ -Dunford-Pettis, if and only if $| T (x_{n}) | \to 0$ for $σ (E, E^{'})$ for every...

On coincidence of Pettis and McShane integrability

Marián J. Fabián (2015)

Czechoslovak Mathematical Journal

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R. Deville and J. Rodríguez proved that, for every Hilbert generated space $X$ , every Pettis integrable function $f : [0, 1] \to X$ is McShane integrable. R. Avilés, G. Plebanek, and J. Rodríguez constructed a weakly compactly generated Banach space $X$ and a scalarly null (hence Pettis integrable) function from $[0, 1]$ into $X$ , which was not McShane integrable. We study here the mechanism behind the McShane integrability of scalarly negligible functions from $[0, 1]$ (mostly) into $C (K)$ spaces. We focus in more detail on...

A note on weakly-supplemented subgroups of finite groups

Hong Pan (2018)

Czechoslovak Mathematical Journal

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A subgroup $H$ of a finite group $G$ is weakly-supplemented in $G$ if there exists a proper subgroup $K$ of $G$ such that $G = H K$ . In the paper, we extend one main result of Kong and Liu (2014).

Radon-Nikodym property

Surjit Singh Khurana (2017)

Commentationes Mathematicae Universitatis Carolinae

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For a Banach space $E$ and a probability space $(X, 𝒜, λ)$ , a new proof is given that a measure $μ : 𝒜 \to E$ , with $μ ≪ λ$ , has RN derivative with respect to $λ$ iff there is a compact or a weakly compact $C \subset E$ such that ${| μ |}_{C} : 𝒜 \to [0, \infty]$ is a finite valued countably additive measure. Here we define ${| μ |}_{C} (A) = sup {\sum_{k} | 〈 μ (A_{k}), f_{k} 〉 |}$ where ${A_{k}}$ is a finite disjoint collection of elements from $𝒜$ , each contained in $A$ , and ${f_{k}} \subset E^{'}$ satisfies ${sup}_{k} | f_{k} (C) | \leq 1$ . Then the result is extended to the case when $E$ is a Frechet space.