-spaces and -spaces
Nguyen To Nhu (1979)
Colloquium Mathematicae
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Nguyen To Nhu (1979)
Colloquium Mathematicae
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Jan Rozendaal, Fedor Sukochev, Anna Tomskova (2016)
Studia Mathematica
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Let X, Y be Banach spaces and let (X,Y) be the space of bounded linear operators from X to Y. We develop the theory of double operator integrals on (X,Y) and apply this theory to obtain commutator estimates of the form for a large class of functions f, where A ∈ (X), B ∈ (Y) are scalar type operators and S ∈ (X,Y). In particular, we establish this estimate for f(t): = |t| and for diagonalizable operators on and for p < q. We also study the estimate above in the setting of Banach...
Luděk Zajíček (2013)
Colloquium Mathematicae
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We construct a Lipschitz function f on X = ℝ ² such that, for each 0 ≠ v ∈ X, the function f is smooth on a.e. line parallel to v and f is Gâteaux non-differentiable at all points of X except a first category set. Consequently, the same holds if X (with dimX > 1) is an arbitrary Banach space and “a.e.” has any usual “measure sense”. This example gives an answer to a natural question concerning the author’s recent study of linearly essentially smooth functions (which generalize essentially...
N. J. Kalton (2011)
Fundamenta Mathematicae
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We show that there is no uniformly continuous selection of the quotient map relative to the unit ball. We use this to construct an answer to a problem of Benyamini and Lindenstrauss; there is a Banach space X such that there is a no Lipschitz retraction of X** onto X; in fact there is no uniformly continuous retraction from onto .
Ferenc Móricz (2009)
Colloquium Mathematicae
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We extend the classical theorems of I. I. Privalov and A. Zygmund from single to multiple conjugate functions in terms of the multiplicative modulus of continuity. A remarkable corollary is that if a function f belongs to the multiplicative Lipschitz class for some and its marginal functions satisfy for some uniformly in the indicated variables , 1 ≤ l ≤ N, then for each choice of with or 1 for 1 ≤ l ≤ N.
Witold Marciszewski (2003)
Studia Mathematica
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We give a characterization of compact spaces K such that the Banach space C(K) is isomorphic to the space c₀(Γ) for some set Γ. As an application we show that there exists an Eberlein compact space K of weight and with the third derived set empty such that the space C(K) is not isomorphic to any c₀(Γ). For this compactum K, the spaces C(K) and are examples of weakly compactly generated (WCG) Banach spaces which are Lipschitz isomorphic but not isomorphic.
Alberto Bressan, Agostino Cortesi (1986)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
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Si dimostra che ogni funzione multivoca lipschitziana con costante di Lipschitz , definita su un sottoinsieme di uno spazio di Hilbert a valori compatti e convessi in , può essere estesa su tutto ad una funzione multivoca lipschitziana con costante minore di 7 nM. In generale, non esistono invece estensioni aventi la stessa costante di Lipschitz .
Ying Xiong, Lifeng Xi (2009)
Studia Mathematica
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This paper studies the geometric structure of graph-directed sets from the point of view of Lipschitz equivalence. It is proved that if and are dust-like graph-directed sets satisfying the transitivity condition, then and are Lipschitz equivalent, and and are quasi-Lipschitz equivalent when they have the same Hausdorff dimension.
Jakub Duda (2009)
Fundamenta Mathematicae
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We find conditions on a real function f:[a,b] → ℝ equivalent to being Lebesgue equivalent to an n-times differentiable function (n ≥ 2); a simple solution in the case n = 2 appeared in an earlier paper. For that purpose, we introduce the notions of and functions, which play analogous rôles for the nth order differentiability to the classical notion of a VBG⁎ function for the first order differentiability, and the classes and (introduced by Preiss and Laczkovich) for Cⁿ smoothness....
Jill Pipher (1992-1993)
Publications mathématiques et informatique de Rennes
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Piotr Niemiec (2009)
Studia Mathematica
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It is proved (independently of the result of Holmes [Fund. Math. 140 (1992)]) that the dual space of the uniform closure of the linear span of the maps x ↦ d(x,a) - d(x,b), where d is the metric of the Urysohn space of diameter r, is (isometrically if r = +∞) isomorphic to the space of equivalence classes of all real-valued Lipschitz maps on . The space of all signed (real-valued) Borel measures on is isometrically embedded in the dual space of and it is shown that the image...
P. Lewis (2001)
Studia Mathematica
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Bessaga and Pełczyński showed that if c₀ embeds in the dual X* of a Banach space X, then ℓ¹ embeds complementably in X, and embeds as a subspace of X*. In this note the Diestel-Faires theorem and techniques of Kalton are used to show that if X is an infinite-dimensional Banach space, Y is an arbitrary Banach space, and c₀ embeds in L(X,Y), then embeds in L(X,Y), and ℓ¹ embeds complementably in . Applications to embeddings of c₀ in various spaces of operators are given.
Alberto Bressan, Agostino Cortesi (1986)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti
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Si dimostra che ogni funzione multivoca lipschitziana con costante di Lipschitz , definita su un sottoinsieme di uno spazio di Hilbert a valori compatti e convessi in , può essere estesa su tutto ad una funzione multivoca lipschitziana con costante minore di 7 nM. In generale, non esistono invece estensioni aventi la stessa costante di Lipschitz .
Vanda Fülöp (2006)
Colloquium Mathematicae
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Denote by the sum of a double sine series with nonnegative coefficients. We present necessary and sufficient coefficient conditions in order that belongs to the two-dimensional multiplicative Lipschitz class Lip(α,β) for some 0 < α ≤ 1 and 0 < β ≤ 1. Our theorems are extensions of the corresponding theorems by Boas for single sine series.
Silvano Delladio (2008)
Bollettino dell'Unione Matematica Italiana
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We prove a result about the rectifiability of class of the set of regular values (in the sense of Clarke) of a Lipschitz map with
Deba P. Sinha, Anil K. Karn (2002)
Studia Mathematica
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For p ≥ 1, a subset K of a Banach space X is said to be relatively p-compact if , where p’ = p/(p-1) and . An operator T ∈ B(X,Y) is said to be p-compact if T(Ball(X)) is relatively p-compact in Y. Similarly, weak p-compactness may be defined by considering . It is proved that T is (weakly) p-compact if and only if T* factors through a subspace of in a particular manner. The normed operator ideals of p-compact operators and of weakly p-compact operators, arising from these factorizations,...
Hôǹg Thái Nguyêñ, Dariusz Pączka (2008)
Banach Center Publications
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Let (Ω,μ) be a measure space, E be an arbitrary separable Banach space, be the dual equipped with the weak* topology, and g:Ω × E → ℝ be a Carathéodory function which is Lipschitz continuous on each ball of E for almost all s ∈ Ω. Put . Consider the integral functional G defined on some non--type Banach space X of measurable functions x: Ω → E. We present several general theorems on sufficient conditions under which any element γ ∈ X* of Clarke’s generalized gradient (multivalued...
Piotr Fijałkowski (2005)
Colloquium Mathematicae
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This paper deals with homeomorphisms F: X → Y, between Banach spaces X and Y, which are of the form where is a continuous (2n+1)-linear operator.
Artur Michalak (2014)
Bulletin of the Polish Academy of Sciences. Mathematics
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We say that a function f from [0,1] to a Banach space X is increasing with respect to E ⊂ X* if x* ∘ f is increasing for every x* ∈ E. A function is separately increasing if it is increasing in each variable separately. We show that if X is a Banach space that does not contain any isomorphic copy of c₀ or such that X* is separable, then for every separately increasing function with respect to any norming subset there exists a separately increasing function such that the sets of...
E. Serrano, C. Piñeiro, J. M. Delgado (2007)
Studia Mathematica
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Let X and Y be Banach spaces. A subset M of (X,Y) (the vector space of all compact operators from X into Y endowed with the operator norm) is said to be equicompact if every bounded sequence (xₙ) in X has a subsequence such that is uniformly convergent for T ∈ M. We study the relationship between this concept and the notion of uniformly completely continuous set and give some applications. Among other results, we obtain a generalization of the classical Ascoli theorem and a compactness...
Elói Medina Galego (2011)
Colloquium Mathematicae
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We classify, up to isomorphism, the spaces of compact operators (E,F), where E and F are the Banach spaces of all continuous functions defined on the compact spaces , the topological products of Cantor cubes and intervals of ordinal numbers [0,α].
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 to a quasi-Banach space ℬ if and only if sup: a is an infinitely differentiable (p,q,s)-atom of < ∞, where the (p,q,s)-atom of is as defined by Han, Paluszyński and Weiss.
Naotsugu Chinen (2015)
Commentationes Mathematicae Universitatis Carolinae
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By , , we denote the -th symmetric product of a metric space as the space of the non-empty finite subsets of with at most elements endowed with the Hausdorff metric . In this paper we shall describe that every isometry from the -th symmetric product into itself is induced by some isometry from into itself, where is either the Euclidean space or the sphere with the usual metrics. Moreover, we study the -th symmetric product of the Euclidean space up to bi-Lipschitz equivalence...
Stanisław Szufla (1989)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti
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For the integral equation (1) below we prove the existence on an interval of a solution with values in a Banach space , belonging to the class , . Further, the set of solutions is shown to be a compact one in the sense of Aronszajn.
Dumitru Popa (2013)
Studia Mathematica
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Let 1 ≤ p < ∞, be a sequence of Banach spaces and the coresponding vector valued sequence space. Let , be two sequences of Banach spaces, , Vₙ: Xₙ → Yₙ, a sequence of bounded linear operators and 1 ≤ p,q < ∞. We define the multiplication operator by . We give necessary and sufficient conditions for to be 2-summing when (p,q) is one of the couples (1,2), (2,1), (2,2), (1,1), (p,1), (p,2), (2,p), (1,p), (p,q); in the last case 1 < p < 2, 1 < q < ∞. ...
Constantin Costara, Dušan Repovš (2010)
Studia Mathematica
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We prove that if F is a Lipschitz map from the set of all complex n × n matrices into itself with F(0) = 0 such that given any x and y we know that F(x) - F(y) and x-y have at least one common eigenvalue, then either or for all x, for some invertible n × n matrix u. We arrive at the same conclusion by supposing F to be of class ¹ on a domain in ℳₙ containing the null matrix, instead of Lipschitz. We also prove that if F is of class ¹ on a domain containing the null matrix satisfying...
Rogério Augusto dos Santos Fajardo (2010)
Bulletin of the Polish Academy of Sciences. Mathematics
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We construct, under Axiom ♢, a family of indecomposable Banach spaces with few operators such that every operator from into is weakly compact, for all ξ ≠ η. In particular, these spaces are pairwise essentially incomparable. Assuming no additional set-theoretic axiom, we obtain this result with size instead of .
Benedetto Silvestri
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The work is dedicated to investigating a limiting procedure for extending “local” integral operator equalities to “global” ones in the sense explained below, and to applying it to obtaining generalizations of the Newton-Leibniz formula for operator-valued functions for a wide class of unbounded operators. The integral equalities considered have the form . (1) They involve functions of the kind , where X is a general locally compact space, F runs over a suitable class of Banach subspaces...
Eva Matoušková, Simeon Reich (2003)
Studia Mathematica
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A Banach space X is reflexive if and only if every bounded sequence xₙ in X contains a norm attaining subsequence. This means that it contains a subsequence for which is attained at some f in the dual unit sphere . A Banach space X is not reflexive if and only if it contains a normalized sequence xₙ with the property that for every , there exists such that . Combining this with a result of Shafrir, we conclude that every infinite-dimensional Banach space contains an unbounded...
Evgeniy Pustylnik (2001)
Studia Mathematica
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We give a full solution of the following problems concerning the spaces : (i) to what extent two functions φ and ψ should be different in order to ensure that for any nontrivial Banach couple X⃗; (ii) when an embedding can (or cannot) be dense; (iii) which Banach space can be regarded as an -space for some (unknown beforehand) Banach couple X⃗.
Tuomas Hytönen, Assaf Naor (2013)
Studia Mathematica
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Given a Banach space X, for n ∈ ℕ and p ∈ (1,∞) we investigate the smallest constant ∈ (0,∞) for which every n-tuple of functions f₁,...,fₙ: -1,1ⁿ → X satisfies , where μ is the uniform probability measure on the discrete hypercube -1,1ⁿ, and and are the hypercube partial derivatives and the hypercube Laplacian, respectively. Denoting this constant by , we show that for every Banach space (X,||·||). This extends the classical Pisier inequality, which corresponds to the special...
Catherine Cabuzel, Alain Pietrus (2007)
Applicationes Mathematicae
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We prove the existence of a sequence satisfying , where f is a function whose second order Fréchet derivative ∇²f satifies a center-Hölder condition and F is a set-valued map from a Banach space X to the subsets of a Banach space Y. We show that the convergence of this method is superquadratic.
Charles E. C Cleaver (1973)
Colloquium Mathematicae
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Grzegorz Bartuzel, Andrzej Fryszkowski (2014)
Banach Center Publications
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In the paper we give an analogue of the Filippov Lemma for the fourth order differential inclusions y = y”” - (A² + B²)y” + A²B²y ∈ F(t,y), (*) with the initial conditions y(0) = y’(0) = y”(0) = y”’(0) = 0, (**) where the matrices are commutative and the multifunction is Lipschitz continuous in y with a t-independent constant l < ||A||²||B||². Main theorem. Assume that y₀ ∈ W4,1 a.e. in [0,1], where p₀ ∈ L¹[0,1]. Then there exists a solution y ∈ W4,1 of (*)...
Thomas Vils Pedersen (2004)
Studia Mathematica
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For 0 < γ ≤ 1, let be the big Lipschitz algebra of functions analytic on the open unit disc which satisfy a Lipschitz condition of order γ on ̅. For a closed set E on the unit circle and an inner function Q, let be the closed ideal in consisting of those functions for which (i) f = 0 on E, (ii) as d(z,E),d(w,E) → 0, (iii) . Also, for a closed ideal I in , let = z ∈ : f(z) = 0 for every f ∈ I and let be the greatest common divisor of the inner parts of non-zero functions...
J. Marshall Ash, Hajrudin Fejzić (2005)
Studia Mathematica
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Let n be a nonnegative integer and let u ∈ (n,n+1]. We say that f is u-times Peano bounded in the approximate (resp. , 1 ≤ p ≤ ∞) sense at if there are numbers , |α| ≤ n, such that is in the approximate (resp. ) sense as h → 0. Suppose f is u-times Peano bounded in either the approximate or sense at each point of a bounded measurable set E. Then for every ε > 0 there is a perfect set Π ⊂ E and a smooth function g such that the Lebesgue measure of E∖Π is less than ε and...