Examples and applications of the density of strongly norm attaining Lipschitz maps

Chiclana, Rafael; García-Lirola, Luis C.; Martı́n, Miguel; Zoca, Abraham Rueda

doi:10.4171/rmi/1253

Cited by 20 publications

(25 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Henceforth we will say that we have strong density for M and Y if the answer is affirmative. Several papers studying this problem have appeared (see [10,Section 7], [3], [6]). Analyzing those results (which we will briefly comment on), it is possible to extract a common idea in them: a Lipschitz map is identified with a bounded linear operator between Banach spaces.…”

Section: Introductionmentioning

confidence: 99%

“…Known results. As already mentioned, all these questions have been studied before in the papers [3,6,7,10,13], among others. Let us present some of the known results concerning these types of density.…”

Section: Introductionmentioning

confidence: 99%

“…We refer to [3,Section 3], where these sufficient conditions and the relationship between them are discussed. Also, we refer to [6,Theorem 2.5], where it is shown that the Radon-Nikodym property and having a uniformly strongly exposed 1-norming set of molecules are not necessary for strong density, and they are distinct properties.…”

Section: Introductionmentioning

confidence: 99%

“…Later, it was generalized to length spaces (see [3,Theorem 2.2]), where the lack of strongly exposed points of B F (M ) seems to be essential in the proofs. However, if we consider T = {x ∈ R 2 : x 2 = 1} endowed with the Euclidean metric, then the unit ball of F(T) is generated by its strongly exposed molecules (in fact, every molecule is strongly exposed), but [6,Theorem 2.1] shows that LipSNA(T, R) is not dense in Lip 0 (T, R).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Some stability properties for the Bishop–Phelps–Bollobás property for Lipschitz maps

Chiclana

Martı́n

2022

Studia Math.

Self Cite

View full text Add to dashboard Cite

We study the stability behavior of the Bishop-Phelps-Bollobás property for Lipschitz maps (Lip-BPB property). This property is a Lipschitz version of the classical Bishop-Phelps-Bollobás property and deals with the possibility of approximating a Lipschitz map that almost attains its (Lipschitz) norm at a pair of distinct points by a Lipschitz map attaining its norm at a pair of distinct points (relatively) very close to the previous one. We first study the stability of this property under the (metric) sum of the domain spaces. Next, we study when it is possible to pass the Lip-BPB property from scalar functions to some vector-valued maps, getting some positive results related to the notions of Γ -flat operators and ACK structure. We get sharper results for the case of Lipschitz compact maps. The behavior of the property with respect to absolute sums of the target space is also studied. We also get results similar to the above for the density of strongly norm attaining Lipschitz maps and of Lipschitz compact maps.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Some stability properties for the Bishop–Phelps–Bollobás property for Lipschitz maps

Chiclana

Martı́n

2022

Studia Math.

Self Cite

View full text Add to dashboard Cite

show abstract

“…In fact, this theory has been widely extended to different contexts besides linear functionals. Indeed, among others, some authors considered it in the context of linear operators (see [13,26,28,31,37,39,40]); others studied norm-attaining bilinear mappings (see [4,8,18]); and more recently several problems on norm-attainment of homogeneous polynomials and Lipschitz maps were considered (see [5,9] and [15,16,17], respectively). In the context of homomorphisms on Banach lattices, we should highlight the recent paper [34], where a James type theorem was proved for positive linear functionals on some Banach lattices (see [34, §6]).…”

Section: Introductionmentioning

confidence: 99%

Norm-attaining lattice homomorphisms

et al. 2021

Self Cite

View full text Add to dashboard Cite

In this paper we study the structure of the set Hom(X, R) of all lattice homomorphisms from a Banach lattice X into R. Using the relation among lattice homomorphisms and disjoint families, we prove that the topological dual of the free Banach lattice F BL(A) generated by a set A contains a disjoint family of cardinality 2 |A| , answering a question of B. de Pagter and A. W. Wickstead. We also deal with norm-attaining lattice homomorphisms. For classical Banach lattices, as c 0 , L p -, and C(K)-spaces, every lattice homomorphism on it attains its norm, which shows, in particular, that there is no James theorem for this class of functions. We prove that, indeed, every lattice homomorphism on X and C(K, X) attains its norm whenever X has order continuous norm. On the other hand, we provide what seems to be the first example in the literature of a lattice homomorphism which does not attain its norm. In general, we study the existence and characterization of lattice homomorphisms not attaining their norm in free Banach lattices. As a consequence, it is shown that no Bishop-Phelps type theorem holds true in the Banach lattice setting, i.e. not every lattice homomorphism can be approximated by norm-attaining lattice homomorphisms.

show abstract