Determining c₀ in C(𝒦) spaces

S. A. Argyros; V. Kanellopoulos

Displaying similar documents to “Determining c₀ in C(𝒦) spaces”

Weakly null sequences with upper estimates

Daniel Freeman (2008)

Studia Mathematica

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We prove that if $(v_{i})$ is a seminormalized basic sequence and X is a Banach space such that every normalized weakly null sequence in X has a subsequence that is dominated by $(v_{i})$ , then there exists a uniform constant C ≥ 1 such that every normalized weakly null sequence in X has a subsequence that is C-dominated by $(v_{i})$ . This extends a result of Knaust and Odell, who proved this for the cases in which $(v_{i})$ is the standard basis for $ℓ_{p}$ or c₀.

Dieudonné operators on the space of Bochner integrable functions

Marian Nowak (2011)

Banach Center Publications

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A bounded linear operator between Banach spaces is called a Dieudonné operator ( = weakly completely continuous operator) if it maps weakly Cauchy sequences to weakly convergent sequences. Let (Ω,Σ,μ) be a finite measure space, and let X and Y be Banach spaces. We study Dieudonné operators T: L¹(X) → Y. Let $i_{\infty} : L^{\infty} (X) \to L ¹ (X)$ stand for the canonical injection. We show that if X is almost reflexive and T: L¹(X) → Y is a Dieudonné operator, then $T \circ i_{\infty} : L^{\infty} (X) \to Y$ is a weakly compact operator. Moreover, we obtain that...

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...

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...

Extension and lifting of weakly continuous polynomials

Raffaella Cilia, Joaquín M. Gutiérrez (2005)

Studia Mathematica

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We show that a Banach space X is an ℒ₁-space (respectively, an $ℒ_{\infty}$ -space) if and only if it has the lifting (respectively, the extension) property for polynomials which are weakly continuous on bounded sets. We also prove that X is an ℒ₁-space if and only if the space $_{w b} (^{m} X)$ of m-homogeneous scalar-valued polynomials on X which are weakly continuous on bounded sets is an $ℒ_{\infty}$ -space.

On projectional skeletons in Vašák spaces

Ondřej F. K. Kalenda (2017)

Commentationes Mathematicae Universitatis Carolinae

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We provide an alternative proof of the theorem saying that any Vašák (or, weakly countably determined) Banach space admits a full $1$ -projectional skeleton. The proof is done with the use of the method of elementary submodels and is comparably simple as the proof given by W. Kubiś (2009) in case of weakly compactly generated spaces.

The Lindelöf property in Banach spaces

B. Cascales, I. Namioka, J. Orihuela (2003)

Studia Mathematica

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A topological space (T,τ) is said to be fragmented by a metric d on T if each non-empty subset of T has non-empty relatively open subsets of arbitrarily small d-diameter. The basic theorem of the present paper is the following. Let (M,ϱ) be a metric space with ϱ bounded and let D be an arbitrary index set. Then for a compact subset K of the product space $M^{D}$ the following four conditions are equivalent: (i) K is fragmented by $d_{D}$ , where, for each S ⊂ D, $d_{S} (x, y) = s u p ϱ (x (t), y (t)) : t \in S$ . (ii) For each countable subset...

Diagonalization in proof complexity

Jan Krajíček (2004)

Fundamenta Mathematicae

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We study diagonalization in the context of implicit proofs of [10]. We prove that at least one of the following three conjectures is true: ∙ There is a function f: 0,1* → 0,1 computable in that has circuit complexity $2^{Ω (n)}$ . ∙ ≠ co . ∙ There is no p-optimal propositional proof system. We note that a variant of the statement (either ≠ co or ∩ co contains a function $2^{Ω (n)}$ hard on average) seems to have a bearing on the existence of good proof complexity generators. In particular, we prove that...

On operators which factor through $l_{p}$ or c₀

Bentuo Zheng (2006)

Studia Mathematica

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Let 1 < p < ∞. Let X be a subspace of a space Z with a shrinking F.D.D. (Eₙ) which satisfies a block lower-p estimate. Then any bounded linear operator T from X which satisfies an upper-(C,p)-tree estimate factors through a subspace of ${(\sum F ₙ)}_{l_{p}}$ , where (Fₙ) is a blocking of (Eₙ). In particular, we prove that an operator from $L_{p}$ (2 < p < ∞) satisfies an upper-(C,p)-tree estimate if and only if it factors through $l_{p}$ . This gives an answer to a question of W. B. Johnson. We also prove...

Characterizing matrices with $𝐗$ -simple image eigenspace in max-min semiring

Ján Plavka, Sergeĭ Sergeev (2016)

Kybernetika

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A matrix $A$ is said to have $𝐗$ -simple image eigenspace if any eigenvector $x$ belonging to the interval $𝐗 = {x : \underset{̲}{x} \leq x \leq \bar{x}}$ is the unique solution of the system $A \otimes y = x$ in $𝐗$ . The main result of this paper is a combinatorial characterization of such matrices in the linear algebra over max-min (fuzzy) semiring. The characterized property is related to and motivated by the general development of tropical linear algebra and interval analysis, as well as the notions of simple image set and weak robustness (or weak stability)...

On families of weakly dependent random variables

Tomasz Łuczak (2011)

Banach Center Publications

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Let $ₙ^{(k)}$ be a family of random independent k-element subsets of [n] = 1,2,...,n and let $(ₙ^{(k)}, ℓ) = ₙ^{(k)} (ℓ)$ denote a family of ℓ-element subsets of [n] such that the event that S belongs to $ₙ^{(k)} (ℓ)$ depends only on the edges of $ₙ^{(k)}$ contained in S. Then, the edges of $ₙ^{(k)} (ℓ)$ are ’weakly dependent’, say, the events that two given subsets S and T are in $ₙ^{(k)} (ℓ)$ are independent for vast majority of pairs S and T. In the paper we present some results on the structure of weakly dependent families of subsets obtained in this way. We...

Weakly precompact operators on $C_{b} (X, E)$ with the strict topology

Juliusz Stochmal (2016)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let $C_{b} (X, E)$ be the space of all E-valued bounded continuous functions on X, equipped with the strict topology β. We study weakly precompact operators $T : C_{b} (X, E) \to F$ . In particular, we show that if X is a paracompact k-space and E contains no isomorphic copy of l¹, then every strongly bounded operator $T : C_{b} (X, E) \to F$ is weakly precompact.

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).

Weakly P-saturated graphs

Mieczysław Borowiecki, Elżbieta Sidorowicz (2002)

Discussiones Mathematicae Graph Theory

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For a hereditary property let $k_{} (G)$ denote the number of forbidden subgraphs contained in G. A graph G is said to be weakly -saturated, if G has the property and there is a sequence of edges of G̅, say $e ₁, e ₂, . . ., e_{l}$ , such that the chain of graphs $G = G ₀ \subset G_{0} + e ₁ \subset G ₁ + e ₂ \subset . . . \subset G_{l - 1} + e_{l} = G_{l} = K_{n} (G_{i + 1} = G_{i} + e_{i + 1})$ has the following property: $k_{} (G_{i + 1}) > k_{} (G_{i})$ , 0 ≤ i ≤ l-1. In this paper we shall investigate some properties of weakly saturated graphs. We will find upper bound for the minimum number of edges of weakly ₖ-saturated graphs of order n. We shall determine the number wsat(n,)...

On the condition of Λ-convexity in some problems of weak continuity and weak lower semicontinuity

Agnieszka Kałamajska (2001)

Colloquium Mathematicae

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We study the functional $I_{f} (u) = \int_{Ω} f (u (x)) d x$ , where u=(u₁, ..., uₘ) and each $u_{j}$ is constant along some subspace $W_{j}$ of ℝⁿ. We show that if intersections of the $W_{j}$ ’s satisfy a certain condition then $I_{f}$ is weakly lower semicontinuous if and only if f is Λ-convex (see Definition 1.1 and Theorem 1.1). We also give a necessary and sufficient condition on ${W_{j}}_{j = 1, . . ., m}$ to have the equivalence: $I_{f}$ is weakly continuous if and only if f is Λ-affine.

A note on weakly-supplemented subgroups and the solvability of finite groups

Xin Liang, Baiyan Xu (2022)

Czechoslovak Mathematical Journal

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Suppose that $G$ is a finite group and $H$ is a subgroup of $G$ . The subgroup $H$ is said to be weakly-supplemented in $G$ if there exists a proper subgroup $K$ of $G$ such that $G = H K$ . In this note, by using the weakly-supplemented subgroups, we point out several mistakes in the proof of Theorem 1.2 of Q. Zhou (2019) and give a counterexample.

On reduction of two-parameter prediction problems

J. Friedrich, L. Klotz, M. Riedel (1995)

Studia Mathematica

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We present a general method for the extension of results about linear prediction for q-variate weakly stationary processes on a separable locally compact abelian group $G_{2}$ (whose dual is a Polish space) with known values of the processes on a separable subset $S_{2} \subseteq G_{2}$ to results for weakly stationary processes on $G_{1} \times G_{2}$ with observed values on $G_{1} \times S_{2}$ . In particular, the method is applied to obtain new proofs of some well-known results of Ze Pei Jiang.

James boundaries and σ-fragmented selectors

B. Cascales, M. Muñoz, J. Orihuela (2008)

Studia Mathematica

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We study the boundary structure for w*-compact subsets of dual Banach spaces. To be more precise, for a Banach space X, 0 < ϵ < 1 and a subset T of the dual space X* such that ⋃ B(t,ϵ): t ∈ T contains a James boundary for $B_{X *}$ we study different kinds of conditions on T, besides T being countable, which ensure that $X * = {\bar{s p a n T}}^{| | \cdot | |}$ . (SP) We analyze two different non-separable cases where the equality (SP) holds: (a) if $J : X \to 2^{B_{X *}}$ is the duality mapping and there exists a σ-fragmented map f: X → X* such that...

On weakly-supplemented subgroups of finite groups

Qingjun Kong (2019)

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 this paper, some interesting results with weakly-supplemented minimal subgroups to a smaller subgroup of $G$ are obtained.

Lower semicontinuous envelopes in $W^{1, 1} \times L^{p}$

Ana Margarida Ribeiro, Elvira Zappale (2014)

Banach Center Publications

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The lower semicontinuity of functionals of the type $\int_{Ω} f (x, u, v, \nabla u) d x$ with respect to the $(W^{1, 1} \times L^{p})$ -weak* topology is studied. Moreover, in absence of lower semicontinuity, an integral representation in $W^{1, 1} \times L^{p}$ for the lower semicontinuous envelope is also provided.

On weakly $θ$ -continuous functions

F. Commaroto, T. Noiri (1986)

Matematički Vesnik

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An elementary proof of Marcellini Sbordone semicontinuity theorem

Tomáš G. Roskovec, Filip Soudský (2023)

Kybernetika

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The weak lower semicontinuity of the functional $F (u) = \int_{Ω} f (x, u, \nabla u) d x$ is a classical topic that was studied thoroughly. It was shown that if the function $f$ is continuous and convex in the last variable, the functional is sequentially weakly lower semicontinuous on $W^{1, p} (Ω)$ . However, the known proofs use advanced instruments of real and functional analysis. Our aim here is to present a proof understandable even for students familiar only with the elementary measure theory.

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...