Coorbit space theory for quasi-Banach spaces

Holger Rauhut

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Displaying similar documents to “Coorbit space theory for quasi-Banach spaces”

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.

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.

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 ().

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.

Some remarks on quasi-Cohen sets

Pascal Lefèvre, Daniel Li (2001)

Colloquium Mathematicae

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We are interested in Banach space geometry characterizations of quasi-Cohen sets. For example, it turns out that they are exactly the subsets E of the dual of an abelian compact group G such that the canonical injection $C (G) / C_{E^{c}} (G) ↪ L ²_{E} (G)$ is a 2-summing operator. This easily yields an extension of a result due to S. Kwapień and A. Pełczyński. We also investigate some properties of translation invariant quotients of L¹ which are isomorphic to subspaces of L¹.

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 < ∞.

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...

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,...

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...

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.

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.

The Dual of a Non-reflexive L-embedded Banach Space Contains $l^{\infty}$ Isometrically

Hermann Pfitzner (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

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A Banach space is said to be L-embedded if it is complemented in its bidual in such a way that the norm between the two complementary subspaces is additive. We prove that the dual of a non-reflexive L-embedded Banach space contains $l^{\infty}$ isometrically.

Estimation of the Szlenk index of Banach spaces via Schreier spaces

Ryan Causey (2013)

Studia Mathematica

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For each ordinal α < ω₁, we prove the existence of a Banach space with a basis and Szlenk index $ω^{α + 1}$ which is universal for the class of separable Banach spaces with Szlenk index not exceeding $ω^{α}$ . Our proof involves developing a characterization of which Banach spaces embed into spaces with an FDD with upper Schreier space estimates.

An approximation property with respect to an operator ideal

Juan Manuel Delgado, Cándido Piñeiro (2013)

Studia Mathematica

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Given an operator ideal , we say that a Banach space X has the approximation property with respect to if T belongs to ${\bar{S \circ T : S \in ℱ (X)}}^{τ_{c}}$ for every Banach space Y and every T ∈ (Y,X), $τ_{c}$ being the topology of uniform convergence on compact sets. We present several characterizations of this type of approximation property. It is shown that some of the existing approximation properties in the literature may be included in this setting.

On the compact approximation property

Vegard Lima, Åsvald Lima, Olav Nygaard (2004)

Studia Mathematica

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We show that a Banach space X has the compact approximation property if and only if for every Banach space Y and every weakly compact operator T: Y → X, the space = S ∘ T: S compact operator on X is an ideal in = span(,T) if and only if for every Banach space Y and every weakly compact operator T: Y → X, there is a net $(S_{γ})$ of compact operators on X such that $s u p_{γ} | | S_{γ} T | | \leq | | T | |$ and $S_{γ} \to I_{X}$ in the strong operator topology. Similar results for dual spaces are also proved.

Schauder bases and the bounded approximation property in separable Banach spaces

Jorge Mujica, Daniela M. Vieira (2010)

Studia Mathematica

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Let E be a separable Banach space with the λ-bounded approximation property. We show that for each ϵ > 0 there is a Banach space F with a Schauder basis such that E is isometrically isomorphic to a 1-complemented subspace of F and, moreover, the sequence (Tₙ) of canonical projections in F has the properties $s u p_{n \in ℕ} | | T ₙ | | \leq λ + ϵ$ and $l i m s u p_{n \to \infty} | | T ₙ | | \leq λ$ . This is a sharp quantitative version of a classical result obtained independently by Pełczyński and by Johnson, Rosenthal and Zippin.

A note on extensions of Pełczyński's decomposition method in Banach spaces

Elói Medina Galego (2007)

Studia Mathematica

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Let X,Y,A and B be Banach spaces such that X is isomorphic to Y ⊕ A and Y is isomorphic to X ⊕ B. In 1996, W. T. Gowers solved the Schroeder-Bernstein problem for Banach spaces by showing that X is not necessarily isomorphic to Y. In the present paper, we give a necessary and sufficient condition on sextuples (p,q,r,s,u,v) in ℕ with p + q ≥ 2, r + s ≥ 1 and u, v ∈ ℕ* for X to be isomorphic to Y whenever these spaces satisfy the following decomposition scheme: ⎧ $X^{u} \sim X^{p} \oplus Y^{q}$ , ⎨ ⎩ $Y^{v} \sim A^{r} \oplus B^{s}$ . Namely, Ω =...

Projections from $L (X, Y)$ onto $K (X, Y)$

Kamil John (2000)

Commentationes Mathematicae Universitatis Carolinae

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Generalization of certain results in [Sap] and simplification of the proofs are given. We observe e.g.: Let $X$ and $Y$ be Banach spaces such that $X$ is weakly compactly generated Asplund space and $X^{*}$ has the approximation property (respectively $Y$ is weakly compactly generated Asplund space and $Y^{*}$ has the approximation property). Suppose that $L (X, Y) \neq K (X, Y)$ and let $1 < λ < 2$ . Then $X$ (respectively $Y$ ) can be equivalently renormed so that any projection $P$ of $L (X, Y)$ onto $K (X, Y)$ has the sup-norm greater or equal to $λ$ . ...

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.

On the existence of non-linear frames

Shah Jahan, Varinder Kumar, S.K. Kaushik (2017)

Archivum Mathematicum

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A stronger version of the notion of frame in Banach space called Strong Retro Banach frame (SRBF) is defined and studied. It has been proved that if $𝒳$ is a Banach space such that $𝒳^{*}$ has a SRBF, then $𝒳$ has a Bi-Banach frame with some geometric property. Also, it has been proved that if a Banach space $𝒳$ has an approximative Schauder frame, then $𝒳^{*}$ has a SRBF. Finally, the existence of a non-linear SRBF in the conjugate of a separable Banach space has been proved.

Multiplying balls in the space of continuous functions on [0,1]

Marek Balcerzak, Artur Wachowicz, Władysław Wilczyński (2005)

Studia Mathematica

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Let C denote the Banach space of real-valued continuous functions on [0,1]. Let Φ: C × C → C. If Φ ∈ +, min, max then Φ is an open mapping but the multiplication Φ = · is not open. For an open ball B(f,r) in C let B²(f,r) = B(f,r)·B(f,r). Then f² ∈ Int B²(f,r) for all r > 0 if and only if either f ≥ 0 on [0,1] or f ≤ 0 on [0,1]. Another result states that Int(B₁·B₂) ≠ ∅ for any two balls B₁ and B₂ in C. We also prove that if Φ ∈ +,·,min,max, then the set $Φ^{- 1} (E)$ is residual whenever E is...

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....