-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.
Amir Sahami, Mohammad R. Omidi, Eghbal Ghaderi, Hamzeh Zangeneh (2020)
Commentationes Mathematicae Universitatis Carolinae
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We study the structure of Lipschitz algebras under the notions of approximate biflatness and Johnson pseudo-contractibility. We show that for a compact metric space , the Lipschitz algebras and are approximately biflat if and only if is finite, provided that . We give a necessary and sufficient condition that a vector-valued Lipschitz algebras is Johnson pseudo-contractible. We also show that some triangular Banach algebras are not approximately biflat.
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...
Lydia Bouchal, Karima Mebarki, Svetlin Georgiev Georgiev b (2022)
Archivum Mathematicum
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In this paper, by using recent results on fixed point index, we develop a new fixed point theorem of functional type for the sum of two operators where is Lipschitz invertible and a -set contraction. This fixed point theorem is then used to establish a new result on the existence of positive solutions to a non-autonomous second order difference equation.
F. Abtahi, E. Byabani, A. Rejali (2019)
Archivum Mathematicum
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Let be a metric space and . We study homological properties and different types of amenability of Lipschitz algebras and their second duals. Precisely, we first provide some basic properties of Lipschitz algebras, which are important for metric geometry to know how metric properties are reflected in simple properties of Lipschitz functions. Then we show that all of these properties are equivalent to either uniform discreteness or finiteness of . Finally, some results concerning...
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.
Luděk Zajíček (2023)
Commentationes Mathematicae Universitatis Carolinae
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Let be Banach spaces and a real function on . Let be the set of all points at which is partially Fréchet differentiable but is not Fréchet differentiable. Our results imply that if are Asplund spaces and is continuous (respectively Lipschitz) on , then is a first category set (respectively a -upper porous set). We also prove that if , are separable Banach spaces and is a Lipschitz mapping, then there exists a -upper porous set such that is Fréchet differentiable...
Michael Dymond (2017)
Commentationes Mathematicae Universitatis Carolinae
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A subset of is called a universal differentiability set if it contains a point of differentiability of every Lipschitz function . We show that any universal differentiability set contains a ‘kernel’ in which the points of differentiability of each Lipschitz function are dense. We further prove that no universal differentiability set may be decomposed as a countable union of relatively closed, non-universal differentiability sets.
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...
Ioana Ghenciu (2023)
Commentationes Mathematicae Universitatis Carolinae
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We introduce the definition of -limited completely continuous operators, . The question of whether a space of operators has the property that every -limited subset is relative compact when the dual of the domain and the codomain have this property is studied using -limited completely continuous evaluation operators.