A Compilation of Some Well-Known Results in Renorming Theory
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On Quasi Norm Attaining Operators Between Banach Spaces
ON QUASI NORM ATTAINING OPERATORS BETWEEN BANACH SPACES GEUNSU CHOI, YUN SUNG CHOI, MINGU JUNG, AND MIGUEL MART´IN Abstract. We provide a characterization of the Radon-Nikod´ymproperty in terms of the denseness of bounded linear operators which attain their norm in a weak sense, which complement the one given by Bourgain and Huff in the 1970's. To this end, we introduce the following notion: an operator T : X ÝÑ Y between the Banach spaces X and Y is quasi norm attaining if there is a sequence pxnq of norm one elements in X such that pT xnq converges to some u P Y with }u}“}T }. Norm attaining operators in the usual (or strong) sense (i.e. operators for which there is a point in the unit ball where the norm of its image equals the norm of the operator) and also compact operators satisfy this definition. We prove that strong Radon-Nikod´ymoperators can be approximated by quasi norm attaining operators, a result which does not hold for norm attaining operators in the strong sense. This shows that this new notion of quasi norm attainment allows to characterize the Radon-Nikod´ymproperty in terms of denseness of quasi norm attaining operators for both domain and range spaces, completing thus a characterization by Bourgain and Huff in terms of norm attaining operators which is only valid for domain spaces and it is actually false for range spaces (due to a celebrated example by Gowers of 1990). A number of other related results are also included in the paper: we give some positive results on the denseness of norm attaining Lipschitz maps, norm attaining multilinear maps and norm attaining polynomials, characterize both finite dimensionality and reflexivity in terms of quasi norm attaining operators, discuss conditions to obtain that quasi norm attaining operators are actually norm attaining, study the relationship with the norm attainment of the adjoint operator and, finally, present some stability results. -
On Fixed Points of One Class of Operators
⥬ â¨ç÷ âã¤÷ù. .7, ü2 Matematychni Studii. V.7, No.2 517.988 ON FIXED POINTS OF ONE CLASS OF OPERATORS Yu.F. Korobe˘ınik Yu. Korobeinik. On xed points of one class of operators, Matematychni Studii, 7(1997) 187{192. Let T : Q ! Q be a selfmapping of some subset Q of a Banach space E. The operator T does not increase the distance between Q and the origin (the null element E) if 8x 2 Q kT xk ≤ kxk (such operator is called a NID operator). In this paper some sucient conditions for the existence of a non-trivial xed point of a NID operator are obtained. These conditions enable in turn to prove some new results on xed points of nonexpanding operators. Let Q be a subset of a linear normed space (E; k · k). An operator T : Q ! E is said to be nonexpanding or NE operator if 8x1; x2 2 Q kT x1 − T x2k ≤ kx1 − x2k: It is evident that each nonexpanding mapping of Q is continuous in Q. The problem of existence of xed points of nonexpanding operators was investi- gated in many papers (see [1]{[9]). The monograph [10] contains a comparatively detailed survey of essential results on xed points of NE operators obtained in the beginning of the seventies. In particular, it was proved that each NE operator in a closed bounded convex subset of a uniformly convex space has at least one xed point in Q (for the de nition of a uniformly convex space see [10], ch.2, x4 or [11], ch.V, x12). -
A Note on Bi-Contractive Projections on Spaces of Vector Valued Continuous
Concr. Oper. 2018; 5: 42–49 Concrete Operators Research Article Fernanda Botelho* and T.S.S.R.K. Rao A note on bi-contractive projections on spaces of vector valued continuous functions https://doi.org/10.1515/conop-2018-0005 Received October 10, 2018; accepted December 12, 2018. Abstract: This paper concerns the analysis of the structure of bi-contractive projections on spaces of vector valued continuous functions and presents results that extend the characterization of bi-contractive projec- tions given by the rst author. It also includes a partial generalization of these results to ane and vector valued continuous functions from a Choquet simplex into a Hilbert space. Keywords: Contractive projections, Bi-contractive projections, Vector measures, Spaces of vector valued functions MSC: 47B38, 46B04, 46E40 1 Introduction We denote by C(Ω, E) the space of all continuous functions dened on a compact Hausdor topological space Ω with values in a Banach space E, endowed with the norm YfY∞ = maxx∈Ω Yf(x)YE. The continuity of a function f ∈ C(Ω, E) is understood to be “in the strong sense", meaning that for every w0 ∈ Ω and > 0 there exists O, an open subset of Ω, containing w0 such that Yf (w) − f(w0)YE < , for every w ∈ O, see [9] or [16]. A contractive projection P ∶ C(Ω, E) → C(Ω, E) is an idempotent bounded linear operator of norm 1. ⊥ A contractive projection is called bi-contractive if its complementary projection P (= I − P) is contractive. In this paper, we extend the characterization given in [6], of bi-contractive projections on C(Ω, E), with Ω a compact metric space, E a Hilbert space that satisfy an additional support disjointness property. -
10. Uniformly Convex Spaces
Researcher, 1(1), 2009, http://www.sciencepub.net, [email protected] Uniformly Convex Spaces Usman, M .A, Hammed, F.A. And Olayiwola, M.O Department Of Mathematical Sciences Olabisi Onabanjo University Ago-Iwoye, Nigeria [email protected] ABSTARACT: This paper depicts the class OF Banach Spaces with normal structure and that they are generally referred to as uniformly convex spaces. A review of properties of a uniformly convex spaces is also examined with a view to show that all non-expansive mapping have a fixed point on this space. With the definition of uniformly convex spaces in mind, we also proved that some spaces are uniformly spaces. [Researcher. 2009;1(1):74-85]. (ISSN: 1553-9865). Keywords: Banach Spaces, Convex spaces, Uniformly Convex Space. INTRODUCTION The importance of Uniformly convex spaces in Applied Mathematics and Functional Analysis, it has developed into area of independent research, where several areas of Mathematics such as Homology theory, Degree theory and Differential Geometry have come to play a very significant role. [1,3,4] Classes of Banach spaces with normal structure are those generally refer to as Uniformly convex spaces. In this paper, we review properties of the space and show that all non- expansive maps have a fixed-point on this space. [2] Let x be a Banach Space. A Branch space x is said to be Uniformly convex if for ε > 0 there exist a ∂ = (ε ) > 0 such that if x, y £ x with //x//=1, //y//=1 and //x-y// ≥ ε , then 1 ()x+ y // ≤ 1 − ∂ // 2 . THEOREM (1.0) Let x = LP (µ) denote the space of measurable function f such that //f// are integrable, endowed with the norm. -
Semi-Indefinite-Inner-Product and Generalized Minkowski Spaces
Semi-indefinite-inner-product and generalized Minkowski spaces A.G.Horv´ath´ Department of Geometry, Budapest University of Technology and Economics, H-1521 Budapest, Hungary Nov. 3, 2008 Abstract In this paper we parallelly build up the theories of normed linear spaces and of linear spaces with indefinite metric, called also Minkowski spaces for finite dimensions in the literature. In the first part of this paper we collect the common properties of the semi- and indefinite-inner-products and define the semi-indefinite- inner-product and the corresponding structure, the semi-indefinite-inner- product space. We give a generalized concept of Minkowski space embed- ded in a semi-indefinite-inner-product space using the concept of a new product, that contains the classical cases as special ones. In the second part of this paper we investigate the real, finite dimen- sional generalized Minkowski space and its sphere of radius i. We prove that it can be regarded as a so-called Minkowski-Finsler space and if it is homogeneous one with respect to linear isometries, then the Minkowski- Finsler distance its points can be determined by the Minkowski-product. MSC(2000):46C50, 46C20, 53B40 Keywords: normed linear space, indefinite and semi-definite inner product, arXiv:0901.4872v1 [math.MG] 30 Jan 2009 orthogonality, Finsler space, group of isometries 1 Introduction 1.1 Notation and Terminology concepts without definition: real and complex vector spaces, basis, dimen- sion, direct sum of subspaces, linear and bilinear mapping, quadratic forms, inner (scalar) product, hyperboloid, ellipsoid, hyperbolic space and hyperbolic metric, kernel and rank of a linear mapping. -
Mazur's Intersection Property and a Krein
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 104, Number 1, September 1988 MAZUR'S INTERSECTION PROPERTY AND A KREIN-MILMAN TYPE THEOREM FOR ALMOST ALL CLOSED, CONVEX AND BOUNDED SUBSETS OF A BANACH SPACE PANDO GRIGOROV GEORGIEV (Communicated by William J. Davis) ABSTRACT. Let 2^ (resp. 2^*) be the set of all closed, convex and bounded (resp. «/-compact and convex) subsets of a Banach space E (resp. of its dual E*) furnished with the Hausdorff metric. It is shown that if there exists an equivalent norm || • || in E with dual || • ||* such that (E, \\ ■ \\) has Mazur's intersection property and (¿?*,|| • ||*) has iu*-Mazur's intersection property, then (1) there exists a dense G s subset % of 2^ such that for every Xe% the strongly exposing functionals form a dense G¡ subset of E* ; (2) there exists a dense G¡ subset 2q* of 2^* such that for every X* 6 2q* the it;"-strongly exposing functionals form a dense G¿ subset of E. In particular every X 6 2^6 is the closed convex hull of its strongly exposed points and every X* 6 2q* is the «/-closed convex hull of its «/-strongly exposed points. Let y be the set of all convex, closed, bounded and nonempty subsets of a real Banach space (E, || • ||) and W* be the set of all convex, «/-compact and nonempty subsets of (E*, || ■ ||*) (the dual space of E). The Hausdorff metric between two subsets of E is defined as follows: h(X, Y) = inf {s >0:XcY+eB,YcX + eB}, where X, Y c E and B is the closed unit ball in E: B = {x G E: \\x\\ < 1}. -
Asplund Sets and Metrizability for the Polynomial Topology
ASPLUND SETS AND METRIZABILITY FOR THE POLYNOMIAL TOPOLOGY PABLO GALINDO AND ALEJANDRO MIRALLES Abstract. The theme of this paper is the study of the separability of subspaces of holomorphic functions respect to the convergence over a given set and its connection with the metrizability of the polynomial topology. A notion closely related to this matter is that of Asplund set. Our discussion includes an affirmative answer to a question of Globevnik about interpolating sequences. We also consider the interplay between polynomials and Asplund sets and derive some consequences of it. Among them we obtain a characterization of Radon-Nikod´ym com- position operators on algebras of bounded analytic functions. 1. Introduction and preliminaries In this article we deal firstly with the natural matter of the separability of some algebras of bounded analytic functions defined on complex Banach spaces E with open unit ball BE which may be seen as the natural extensions of the disc algebra. We will prove that the algebra A∞(BE) := {f : BE → C : f continuous, bounded and analytic on BE} endowed with the sup norm is separable if, and only if, E is finite-dimen- sional. This is achieved by answering affirmatively the question, raised by Globevnik [Gl], of the existence in BE of interpolating sequences for A∞(BE) if E a non-reflexive Banach space. The separability of the subalgebra Au(BE) ⊂ A∞(BE) of analytic and uniformly continuous functions on BE is also discussed and we give a sim- ple characterization of it in terms of the metrizability of its spectrum for the polynomial topology, that is, the topology τ(P (E)) of the convergence against the continuous polynomials on E. -
On the Weak* Mazur Intersection Property and Frechet Differentiable Norms on Dense Open Sets
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Bull. Sci. math. 1998, 122, p. 93-105 ON THE WEAK* MAZUR INTERSECTION PROPERTY AND FRECHET DIFFERENTIABLE NORMS ON DENSE OPEN SETS BY J. P. MORENO (*) ABSTRACT.- We prove in this paper the following renorming result: every Banach space with a fundamental biorthogonal system admits an equivalent norm with a dense set of locally uniformly rotund points. Then we obtain new sufficient and necessary conditions for the existence of Frechet differentiable norms and norms with the Mazur (weak* Mazur) intersection property. These conditions are expressed in terms of norms with are Frechet differentiable on dense open sets. 0 Elsevier, Paris 1. Introduction It was MAZUR [14] who first considered the study of those Banach spaces which have the euclidean spaceproperty later called the Muzur intersection property: every bounded closed convex set can be represented as an intersection of closed balls. A systematic study of this topic was initiated by PHELPS [ 181 and continued by SULLIVAN [20]. Later, GILES, GREGORY and SIMS gave some characterisations of this property [.5]. They also (*) Manuscript presentedby G. PISIER, received in March 1996. 1991 Mathematics Subject Classification 46 B 20. .I. P. MORENO, Dpto de Analisis matematico, Facultad de Matematicas. Universidad Complutense de Madrid, E-28040 Madrid (Spain). Partially supported by DGICYT PB 93-0452. BULLETIN DES SCIENCES MATHBMATIQUES - 0007.4497/98/02 0 Elsevier, Paris 94 J. P. MORENO considered, for a dual space, the weak* Mazur intersection property: every bounded weak* closed convex set can be represented as an intersection of closed dual balls. -
Duality Results in Banach and Quasi-Banach Spaces Of
DUALITY RESULTS IN BANACH AND QUASI-BANACH SPACES OF HOMOGENEOUS POLYNOMIALS AND APPLICATIONS VIN´ICIUS V. FAVARO´ AND DANIEL PELLEGRINO Abstract. Spaces of homogeneous polynomials on a Banach space are frequently equipped with quasi- norms instead of norms. In this paper we develop a technique to replace the original quasi-norm by a norm in a dual preserving way, in the sense that the dual of the space with the new norm coincides with the dual of the space with the original quasi-norm. Applications to problems on the existence and approximation of solutions of convolution equations and on hypercyclic convolution operators on spaces of entire functions are provided. Contents 1. Introduction 1 2. Main results 2 3. Prerequisites for the applications 9 3.1. Hypercyclicity results 11 3.2. Existence and approximation results 12 4. Applications 12 4.1. Lorentz nuclear and summing polynomials: the basics 12 4.2. New hypercyclic, existence and approximation results 14 References 22 Mathematics Subject Classifications (2010): 46A20, 46G20, 46A16, 46G25, 47A16. Key words: Banach and quasi-Banach spaces, homogeneous polynomials, entire functions, convolution operators, Lorentz sequence spaces. 1. Introduction arXiv:1503.01079v2 [math.FA] 28 Jul 2016 In the 1960s, many researchers began the study of spaces of holomorphic functions defined on infinite dimensional complex Banach spaces. In this context spaces of n-homogeneous polynomials play a central role in the development of the theory. Several tools, such as topological tensor products and duality theory, are useful and important when we are working with spaces of homogeneous polynomials. Many duality results on spaces of homogeneous polynomials and their applications have appeared in the last decades (see for instance [5, 8, 9, 10, 11, 12, 15, 20, 22, 26, 28, 29, 33, 34, 41, 46] among others). -
ON the STABILITY of the DIFFERENTIAL PROCESS GENERATED by COMPLEX INTERPOLATION Contents 1. Introduction 2 2. Preliminary Result
ON THE STABILITY OF THE DIFFERENTIAL PROCESS GENERATED BY COMPLEX INTERPOLATION JESUS´ M. F. CASTILLO, WILLIAN H. G. CORREA,^ VALENTIN FERENCZI, MANUEL GONZALEZ´ Abstract. We study the stability of the differential process of Rochberg and Weiss associated to an analytic family of Banach spaces obtained using the complex interpolation method for families. In the context of K¨othefunction spaces we complete earlier results of Kalton (who showed that there is global bounded stability for pairs of K¨othespaces) by showing that there is global (bounded) stability for families of up to three K¨othespaces distributed in arcs on the unit sphere while there is no (bounded) stability for families of four or more K¨othespaces. In the context of arbitrary pairs of Banach spaces we present some local stability results and some global isometric stability results. Contents 1. Introduction 2 2. Preliminary results 4 3. Kalton spaces of analytic functions 6 3.1. Derivations, centralizers and twisted sums 6 3.2. Distances and isomorphisms 8 3.3. Bounded stability 10 4. Stability of splitting for K¨othefunction spaces 11 4.1. Stability for pairs of K¨othespaces 12 4.2. Complex interpolation for families 15 4.3. Stability for families of three K¨othespaces 17 4.4. (Bounded) stability fails for families of four K¨othespaces 22 5. Stability of splitting for pairs of Banach spaces 25 5.1. Local bounded stability for coherent pairs 26 5.2. Isometric rigidity of linear derivations for optimal interpolation pairs 28 References 30 2010 Mathematics Subject Classification. 46B70, 46E30, 46M18. Key words and phrases. -
Nonsmooth Sequential Analysis in Asplund Spaces
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 348, Number 4, April 1996 NONSMOOTH SEQUENTIAL ANALYSIS IN ASPLUND SPACES BORIS S. MORDUKHOVICH AND YONGHENG SHAO Abstract. We develop a generalized differentiation theory for nonsmooth functions and sets with nonsmooth boundaries defined in Asplund spaces. This broad subclass of Banach spaces provides a convenient framework for many im- portant applications to optimization, sensitivity, variational inequalities, etc. Our basic normal and subdifferential constructions are related to sequential weak-star limits of Fr´echet normals and subdifferentials. Using a variational approach, we establish a rich calculus for these nonconvex limiting objects which turn out to be minimal among other set-valued differential construc- tions with natural properties. The results obtained provide new developments in infinite dimensional nonsmooth analysis and have useful applications to optimization and the geometry of Banach spaces. 1. Introduction It is well known that nonsmooth functions, sets with nonsmooth boundaries and set-valued mappings appear naturally and frequently in various areas of mathe- matics and applications. We refer the reader to [2, 10, 11, 39, 44, 48, 63] for many examples and motivations, especially related to optimization, stability, variational systems, and control systems. A systematic study of the local behavior of nondiffer- entiable objects is accomplished in the framework of nonsmooth analysis, which has become an active and fruitful area of mathematics, rich with applications. The term “nonsmooth analysis” was coined in the 1970s by Francis Clarke, who performed pioneering work in this area for fairly general nonsmooth objects. The heart of nonsmooth analysis is the generalized differentiation of nonsmooth functions. -
Some Types of Banach Spaces, Hermitian Operators, And
SOME TYPES OF BANACHSPACES, HERMITIAN OPERATORS,AND BADE FUNCTIONAL^1) BY EARL BERKSON Introduction. In [6] and [ll] a general notion of hermitian operator has been developed for arbitrary complex Banach spaces (see §1 below). In terms of this notion, a family of operators on a Banach space is said to be hermitian-equivalent if the operators of this family can be made simultaneously hermitian by equivalent renorming of the underlying space [7]. Let X be a complex Banach space with norm || ||, and let F be a commutative hermitian-equivalent family of operators on X. A norm for X equivalent to || ||, and relative to which the operators of F are hermitian will be called an F-norm. Such families have been studied in [7], where it is shown that if X is a Hilbert space, then there is an F-norm which is also a Hilbert space norm. It is natural to seek other properties which, if enjoyed by X, can be preserved by choosing an F-norm appro- priately. Such an investigation is conducted in this paper. Specifically, we show in §§3 and 4 that if the Banach space X has uniformly Frechet differentiable norm (resp., is uniformly convex), then there is an F-norm which preserves uniform Frechet differentiability (resp., uniform con- vexity). Moreover, we show in §6 that if X = Lp(fi), <*>> p > 1, M a meas- ure, then there is an F-norm which preserves both uniform Frechet differ- entiability and uniform convexity. Our result for'the case where X is uniformly convex enables us to es- tablish in Theorem (5.4) a strong link between the notions of semi-inner- product (see §1) and Bade functional.