Some remarks on quasi-Cohen sets

Pascal Lefèvre; Daniel Li

Displaying similar documents to “Some remarks on quasi-Cohen sets”

Quasi-constricted linear operators on Banach spaces

Eduard Yu. Emel&#039;yanov, Manfred P. H. Wolff (2001)

Studia Mathematica

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Let X be a Banach space over ℂ. The bounded linear operator T on X is called quasi-constricted if the subspace $X ₀ : = x \in X : l i m_{n \to \infty} | | T ⁿ x | | = 0$ is closed and has finite codimension. We show that a power bounded linear operator T ∈ L(X) is quasi-constricted iff it has an attractor A with Hausdorff measure of noncompactness $χ_{| | \cdot | | ₁} (A) < 1$ for some equivalent norm ||·||₁ on X. Moreover, we characterize the essential spectral radius of an arbitrary bounded operator T by quasi-constrictedness of scalar multiples of T. Finally, we prove...

Inclusion Indices of Quasi-Banach Spaces

Fernando Cobos, Luz M. Fernández-Cabrera, Antonio Manzano, Antón Martínez (2007)

Bollettino dell'Unione Matematica Italiana

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We investigate inclusion indices for quasi-Banach spaces. First we consider the case of function spaces on $[0,1]$ , then the sequence case and finally we develop an abstract approach dealing with indices defined by the real interpolation scale gen- erated by a quasi-Banach couple.

Conditionality constants of quasi-greedy bases in super-reflexive Banach spaces

F. Albiac, J. L. Ansorena, G. Garrigós, E. Hernández, M. Raja (2015)

Studia Mathematica

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We show that in a super-reflexive Banach space, the conditionality constants $k_{N} (ℬ)$ of a quasi-greedy basis ℬ grow at most like $O ({(l o g N)}^{1 - ε})$ for some 0 < ε < 1. This extends results by the third-named author and Wojtaszczyk (2014), where this property was shown for quasi-greedy bases in $L_{p}$ for 1 < p < ∞. We also give an example of a quasi-greedy basis ℬ in a reflexive Banach space with $k_{N} (ℬ) \approx l o g N$ .

Quasi-algebraic representability of sets in $R^{n}$ .

W. Waliszewski (1984)

Banach Center Publications

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Fine and quasi connectedness in nonlinear potential theory

David R. Adams, John L. Lewis (1985)

Annales de l'institut Fourier

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If $B_{α, p}$ denotes the Bessel capacity of subsets of Euclidean $n$ -space, $α > 0$ , $1 < p < \infty$ , naturally associated with the space of Bessel potentials of $L^{p}$ -functions, then our principal result is the estimate: for $1 < α p \leq n$ , there is a constant $C = C (α, p, n)$ such that for any set $E$ $min {B_{α, p} (E \cap Q), B_{α, p} (E^{c} \cap Q)} \leq C \cdot B_{α, p} (Q \cap \partial_{f} E)$ for all open cubes $Q$ in $n$ -space. Here $\partial_{f} E$ is the boundary of the $E$ in the $(α, p)$ -fine topology i.e. the smallest topology on $c$ -space that makes the associated $(α, p)$ -linear potentials continuous there. As a consequence,...

On quasi-compactness of operator nets on Banach spaces

Eduard Yu. Emel'yanov (2011)

Studia Mathematica

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The paper introduces a notion of quasi-compact operator net on a Banach space. It is proved that quasi-compactness of a uniform Lotz-Räbiger net ${(T_{λ})}_{λ}$ is equivalent to quasi-compactness of some operator $T_{λ}$ . We prove that strong convergence of a quasi-compact uniform Lotz-Räbiger net implies uniform convergence to a finite-rank projection. Precompactness of operator nets is also investigated.

Norm continuity of weakly quasi-continuous mappings

Alireza Kamel Mirmostafaee (2011)

Colloquium Mathematicae

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Let be the class of Banach spaces X for which every weakly quasi-continuous mapping f: A → X defined on an α-favorable space A is norm continuous at the points of a dense $G_{δ}$ subset of A. We will show that this class is stable under c₀-sums and $ℓ^{p}$ -sums of Banach spaces for 1 ≤ p < ∞.

Isomorphisms of some reflexive algebras

Jiankui Li, Zhidong Pan (2008)

Studia Mathematica

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Suppose ℒ₁ and ℒ₂ are subspace lattices on complex separable Banach spaces X and Y, respectively. We prove that under certain lattice-theoretic conditions every isomorphism from algℒ₁ to algℒ₂ is quasi-spatial; in particular, if a subspace lattice ℒ of a complex separable Banach space X contains a sequence $E_{i}$ such that $(E_{i}) ₋ \neq X$ , $E_{i} \subseteq E_{i + 1}$ , and $⋁_{i = 1}^{\infty} E_{i} = X$ then every automorphism of algℒ is quasi-spatial.

Representation and construction of homogeneous and quasi-homogeneous $n$ -ary aggregation functions

Yong Su, Radko Mesiar (2021)

Kybernetika

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Homogeneity, as one type of invariantness, means that an aggregation function is invariant with respect to multiplication by a constant, and quasi-homogeneity, as a relaxed version, reflects the original output as well as the constant. In this paper, we characterize all homogeneous/quasi-homogeneous $n$ -ary aggregation functions and present several methods to generate new homogeneous/quasi-homogeneous $n$ -ary aggregation functions by aggregation of given ones.

Countable compactness and $p$ -limits

Salvador García-Ferreira, Artur Hideyuki Tomita (2001)

Commentationes Mathematicae Universitatis Carolinae

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For $\emptyset \neq M \subseteq ω^{*}$ , we say that $X$ is quasi $M$ -compact, if for every $f : ω \to X$ there is $p \in M$ such that $\bar{f} (p) \in X$ , where $\bar{f}$ is the Stone-Čech extension of $f$ . In this context, a space $X$ is countably compact iff $X$ is quasi $ω^{*}$ -compact. If $X$ is quasi $M$ -compact and $M$ is either finite or countable discrete in $ω^{*}$ , then all powers of $X$ are countably compact. Assuming $C H$ , we give an example of a countable subset $M \subseteq ω^{*}$ and a quasi $M$ -compact space $X$ whose square is not countably compact, and show that in a model of A. Blass and S. Shelah...

Stability of the Cauchy functional equation in quasi-Banach spaces

Jacek Tabor (2004)

Annales Polonici Mathematici

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Let X be a quasi-Banach space. We prove that there exists K > 0 such that for every function w:ℝ → X satisfying ||w(s+t)-w(s)-w(t)|| ≤ ε(|s|+|t|) for s,t ∈ ℝ, there exists a unique additive function a:ℝ → X such that a(1)=0 and ||w(s)-a(s)-sθ(log₂|s|)|| ≤ Kε|s| for s ∈ ℝ, where θ: ℝ → X is defined by $θ (k) : = w (2^{k}) / 2^{k}$ for k ∈ ℤ and extended in a piecewise linear way over the rest of ℝ.

Quasi-greedy bases and Lebesgue-type inequalities

S. J. Dilworth, M. Soto-Bajo, V. N. Temlyakov (2012)

Studia Mathematica

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We study Lebesgue-type inequalities for greedy approximation with respect to quasi-greedy bases. We mostly concentrate on the $L_{p}$ spaces. The novelty of the paper is in obtaining better Lebesgue-type inequalities under extra assumptions on a quasi-greedy basis than known Lebesgue-type inequalities for quasi-greedy bases. We consider uniformly bounded quasi-greedy bases of $L_{p}$ , 1 < p < ∞, and prove that for such bases an extra multiplier in the Lebesgue-type inequality can be taken...

Boundedness of sublinear operators in Triebel-Lizorkin spaces via atoms

Liguang Liu, Dachun Yang (2009)

Studia Mathematica

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Let s ∈ ℝ, p ∈ (0,1] and q ∈ [p,∞). It is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from the Triebel-Lizorkin space $Ḟ_{p, q}^{s} (ℝ ⁿ)$ to a quasi-Banach space ℬ if and only if sup ${| | T (a) | |}_{ℬ}$ : a is an infinitely differentiable (p,q,s)-atom of $Ḟ_{p, q}^{s} (ℝ ⁿ)$ < ∞, where the (p,q,s)-atom of $Ḟ_{p, q}^{s} (ℝ ⁿ)$ is as defined by Han, Paluszyński and Weiss.

An inconsistency equation involving means

Roman Ger, Tomasz Kochanek (2009)

Colloquium Mathematicae

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We show that any quasi-arithmetic mean $A_{φ}$ and any non-quasi-arithmetic mean M (reasonably regular) are inconsistent in the sense that the only solutions f of both equations $f (M (x, y)) = A_{φ} (f (x), f (y))$ and $f (A_{φ} (x, y)) = M (f (x), f (y))$ are the constant ones.

Some results on quasi-t-dual Baer modules

Rachid Tribak, Yahya Talebi, Mehrab Hosseinpour (2023)

Commentationes Mathematicae Universitatis Carolinae

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Let $R$ be a ring and let $M$ be an $R$ -module with $S = {End}_{R} (M)$ . Consider the preradical $\bar{Z}$ for the category of right $R$ -modules Mod- $R$ introduced by Y. Talebi and N. Vanaja in 2002 and defined by $\bar{Z} (M) = ⋂ {U \leq M : M / U$ is small in its injective hull $}$ . The module $M$ is called quasi-t-dual Baer if $\sum_{ϕ \in ℑ} ϕ ({\bar{Z}}^{2} (M))$ is a direct summand of $M$ for every two-sided ideal $ℑ$ of $S$ , where ${\bar{Z}}^{2} (M) = \bar{Z} (\bar{Z} (M))$ . In this paper, we show that $M$ is quasi-t-dual Baer if and only if ${\bar{Z}}^{2} (M)$ is a direct summand of $M$ and ${\bar{Z}}^{2} (M)$ is a quasi-dual Baer module. It is also shown that any direct...

Coorbit space theory for quasi-Banach spaces

Holger Rauhut (2007)

Studia Mathematica

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We generalize the classical coorbit space theory developed by Feichtinger and Gröchenig to quasi-Banach spaces. As a main result we provide atomic decompositions for coorbit spaces defined with respect to quasi-Banach spaces. These atomic decompositions are used to prove fast convergence rates of best n-term approximation schemes. We apply the abstract theory to time-frequency analysis of modulation spaces $M_{m}^{p, q}$ , 0 < p,q ≤ ∞.

From restricted type to strong type estimates on quasi-Banach rearrangement invariant spaces

María Carro, Leonardo Colzani, Gord Sinnamon (2007)

Studia Mathematica

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Let X be a quasi-Banach rearrangement invariant space and let T be an (ε,δ)-atomic operator for which a restricted type estimate of the form $∥ T χ_{E} ∥_{X} \leq D (| E |)$ for some positive function D and every measurable set E is known. We show that this estimate can be extended to the set of all positive functions f ∈ L¹ such that ${| | f | |}_{\infty} \leq 1$ , in the sense that $∥ T f ∥_{X} \leq D (| | f | | ₁)$ . This inequality allows us to obtain strong type estimates for T on several classes of spaces as soon as some information about the galb of the space X is known....

Generalized weighted quasi-arithmetic means and the Kolmogorov-Nagumo theorem

Janusz Matkowski (2013)

Colloquium Mathematicae

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A generalization of the weighted quasi-arithmetic mean generated by continuous and increasing (decreasing) functions $f ₁, . . ., f_{k} : I \to ℝ$ , k ≥ 2, denoted by $A^{[f ₁, . . ., f_{k}]}$ , is considered. Some properties of $A^{[f ₁, . . ., f_{k}]}$ , including “associativity” assumed in the Kolmogorov-Nagumo theorem, are shown. Convex and affine functions involving this type of means are considered. Invariance of a quasi-arithmetic mean with respect to a special mean-type mapping built of generalized means is applied in solving a functional equation. For...

Uniqueness of unconditional basis of $ℓ_{p} (c ₀)$ and $ℓ_{p} (ℓ ₂)$ , 0 < p < 1

F. Albiac, C. Leránoz (2002)

Studia Mathematica

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We prove that the quasi-Banach spaces $ℓ_{p} (c ₀)$ and $ℓ_{p} (ℓ ₂)$ (0 < p < 1) have a unique unconditional basis up to permutation. Bourgain, Casazza, Lindenstrauss and Tzafriri have previously proved that the same is true for the respective Banach envelopes $ℓ ₁ (c ₀)$ and ℓ₁(ℓ₂). They used duality techniques which are not available in the non-locally convex case.

Compactness and extreme points of the set of quasi-measure extensions of a quasi-measure

Zbigniew Lipecki

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The memoir is based on a series of six papers by the author published over the years 1995-2007. It continues the work of D. Plachky (1970, 1976). It also owes some inspiration, among others, to papers by J. Łoś and E. Marczewski (1949), D. Bierlein and W. J. A. Stich (1989), D. Bogner and R. Denk (1994), and A. Ülger (1996). Let and ℜ be algebras of subsets of a set Ω with ⊂ ℜ. Given a quasi-measure μ on , i.e., μ ∈ ba₊(), we denote by E(μ) the convex set of all quasi-measure extensions...

Bounded elements and spectrum in Banach quasi *-algebras

Camillo Trapani (2006)

Studia Mathematica

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A normal Banach quasi *-algebra (,) has a distinguished Banach *-algebra $_{b}$ consisting of bounded elements of . The latter *-algebra is shown to coincide with the set of elements of having finite spectral radius. If the family () of bounded invariant positive sesquilinear forms on contains sufficiently many elements then the Banach *-algebra of bounded elements can be characterized via a C*-seminorm defined by the elements of ().

Quasi-completeness on the Spaces of Holomorphic Germs

Roberto Luiz Soraggi (1981)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Sia $E$ uno spazio $D F$ riflessivo e sia $K$ un compatto di $E$ . Si dimostra che lo spazio dei germi olomorfi su $K$ , con la topologia naturale, è un limite induttivo regolare e quasi completo purché lo spazio dei germi olomorfi all'origine sia un limite induttivo regolare.