Lipschitz compact operators

Jiménez-Vargas, A.; Sepulcre, Juan Matías; Villegas-Vallecillos, Moisés

doi:10.1016/j.jmaa.2014.02.012

Cited by 45 publications

(45 citation statements)

References 16 publications

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“…Proof. It is shown in [14] that f → f t defines an isometry from Lip(M, X) onto L w * ,w * (X * , Lip(M ))), where f t (x * ) = x * • f . Let f be in lip τ (M, X) and let us prove that f t ∈ K w * ,w (X * , lip τ (M )).…”

Section: And Only If the Following Three Conditions Holdmentioning

confidence: 99%

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On the structure of spaces of vector-valued Lipschitz functions

García-Lirola¹,

Petitjean²,

Zoca³

2017

Studia Math.

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Section: And Only If the Following Three Conditions Holdmentioning

confidence: 99%

“…Furthermore, it is known that, for every metric space M and every Banach space X, Lip(M, X) = L(F(M ), X) (e.g. [14]) and that F(M, X) = F(M ) ⊗ π X (see [3, Proposition 1.1]).…”

Section: Introductionmentioning

confidence: 99%

On the structure of spaces of vector-valued Lipschitz functions

García-Lirola¹,

Petitjean²,

Zoca³

2017

Studia Math.

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“…Observe that if Y is finite-dimensional then every Lipschitz map is indeed a Lipschitz compact map, whereas we cannot say that when X is finite-dimensional. We refer to [7, §8.6] and [15] for background. Now we apply the five definitions of norm attainment to the set of Lipschitz compact maps to get the corresponding norm attaining sets: given Banach spaces X, Y , we write…”

Section: Introductionmentioning

confidence: 99%

Emerging notions of norm attainment for Lipschitz maps between Banach spaces

Choi

Martı́n

2020

Journal of Mathematical Analysis and Applications

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We classify several notions of norm attaining Lipschitz maps which were introduced previously, and present the relations among them in order to verify proper inclusions. We also analyze some results for the sets of Lipschitz maps satisfying each of these properties to be dense or not in Lip 0 (X, Y ). For instance, we characterize a Banach space Y with the Radon-Nikodým property in terms of the denseness of norm attaining Lipschitz maps with values in Y . Further, we introduce a property called the local directional Bishop-Phelps-Bollobás property for Lipschitz compact maps, which extends the one studied previously for scalar-valued functions, and provide some new positive results.Recently, a few papers dealing with alternative types of norm attainment for Lipschitz maps defined on Banach spaces have appeared. Kadets, Martín and Soloviova [16, Definition 4.2] introduced another possible definition called (locally) directionally norm attaining Lipschitz function. On the other hand, Godefroy [14] defined other two ways in which a Lipschitz map can attain its norm. We also refer to section 8.8 of the very recent book [7] for an exposition of the results of the two aforementioned papers [14,16]. Our first aim in this paper is to introduce some variations of these definitions of norm attainment and study the possible denseness of the set of Lipschitz maps attaining each of such norms. We first provide with the definitions used throughout the paper. Definitions 1.2 and 1.3 were first introduced in [14], and Definitions 1.4 and 1.5 were first considered in [16] only for Lipschitz (real-valued) functions, which are easily extensible to the general (vector-valued) Lipschitz maps. We will use the usual notation of B X , S X , X * for the closed unit ball, unit sphere, and topological dual, respectively, of a Banach space X.Definition 1.2 ([14]). We say that f ∈ Lip 0 (X, Y ) attains its norm at x ∈ X through a derivative in the direction e ∈ S X if f (x, e) = lim t→0

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“…The Lipschitz p-summing operators can be seen as the first of a large list of different classes of Lipschitz operators that had been studied in the last years. Most of these classes of Lipschitz operators are obtained as a generalization of linear operators, for instance the finite rank, approximable and compact operators [13], p-nuclear and p-integral operators [7], among many others. The necessity to study this different classes in a general framework, unifying results and the language, leads to the new concept of Banach Lipschitz operator ideal defined in [1, Definition 2.1] and independently in [3,Definition 2.3] (under the name of generic Lipschitz operator Banach ideal).…”

Section: Introductionmentioning

confidence: 99%

Galois connection between Lipschitz and linear operator ideals and minimal Lipschitz operator ideals

Turco

Villafañe

2019

Journal of Functional Analysis

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We establish a relation between Lipschitz operator ideals and linear operator ideals, which fits in the framework of Galois connection between lattices. We use this relationship to give a criterion which allow us to recognize when a Banach Lipschitz operator ideal is of composition type or not. Also, we introduce the concept of minimal Banach Lipschitz operator ideal, which have analogous properties to minimal Banach operator ideals. Also we characterize minimal Banach Lipschitz operator ideals which are of composition type and present examples which are not of this class.

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Lipschitz compact operators

Cited by 45 publications

References 16 publications

On the structure of spaces of vector-valued Lipschitz functions

On the structure of spaces of vector-valued Lipschitz functions

Emerging notions of norm attainment for Lipschitz maps between Banach spaces

Galois connection between Lipschitz and linear operator ideals and minimal Lipschitz operator ideals

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