The Bishop–Phelps–Bollobás Property and Absolute Sums

Choi, Yun Sung; Dantas, Sheldon; Jung, Mingu; Martı́n, Miguel

doi:10.1007/s00009-019-1346-6

Cited by 8 publications

(4 citation statements)

References 29 publications

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“…If an analogous definition is valid for operators T and S belonging to a subspace M ⊆ L(X, Y ), then we say that (X, Y ) has the BPBp for operators from M. There is a vast literature about this topic, and we refer the reader to the already cited [3], to [7,12,13,15], and to the references therein. Let us comment that the mentioned result by Bollobás just says that the pair (X, R) has the BPBp for every Banach space X.…”

Section: Introductionmentioning

confidence: 99%

The Bishop–Phelps–Bollobás property for Lipschitz maps

Chiclana

Martı́n

2019

Nonlinear Analysis

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In this paper, we introduce and study a Lipschitz version of the Bishop-Phelps-Bollobás property (Lip-BPB property). This property deals with the possibility of making a uniformly simultaneous approximation of a Lipschitz map F and a pair of points at which F almost attains its norm by a Lipschitz map G and a pair of points such that G strongly attains its norm at the new pair of points. We first show that if M is a finite pointed metric space and Y is a finite-dimensional Banach space, then the pair (M, Y ) has the Lip-BPB property, and that both finiteness assumptions are needed. Next, we show that if M is a uniformly Gromov concave pointed metric space (i.e. the molecules of M form a set of uniformly strongly exposed points), then (M, Y ) has the Lip-BPB property for every Banach space Y . We further prove that this is the case for finite concave metric spaces, ultrametric spaces, and Hölder metric spaces. The extension of the Lip-BPB property from (M, R) to some Banach spaces Y and some results for compact Lipschitz maps are also discussed.Recently, the problem of deciding for which metric spaces M the set LipSNA(M, Y ) is (norm) dense in Lip 0 (M, Y ) has been studied. We refer the reader to [11], [16], [18], and [21], as references for this study. Let us collect here some of the known results. First, the density does not always hold, as in [21, Example 2.1] it is shown that LipSNA([0, 1], R) is not dense in Lip 0 ([0, 1], R). In fact, this can be generalized to length spaces [11, Theorem 2.2]. On the other hand, it is obvious that LipSNA(M, Y ) = Lip 0 (M, Y ) when M is finite (actually, this fact characterizes finiteness of M ). Besides, LipSNA(M, Y ) is dense in Lip 0 (M, Y ) for every Banach space Y when M is uniformly discrete, or M is countable and compact, or M is a compact Hölder metric space (i.e. M = (N, d θ ) for some metric space (N, d) and 0 < θ < 1), but

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Section: Introductionmentioning

confidence: 99%

The Bishop–Phelps–Bollobás property for Lipschitz maps

Chiclana

Martı́n

2019

Nonlinear Analysis

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“…The proof is an adaptation of corresponding one for the BPBp given in [11]. On the other hand, when the absolute sum is ∞ -or 1 -sum, then it is not needed to divide ε by 3 in the above result since we may adapt the arguments given in [8, Propositions 2.4 and 2.7].…”

Section: Some Stability Resultsmentioning

confidence: 98%

“…Let us comment that we do not know whether the analogous results for the BPBp are true. We send the reader to the very recent paper [11] to see some particular cases in which this is the case. We also do not know whether the density of norm attaining operators passes to one complemented subspaces of the domain space.…”

Section: Some Stability Resultsmentioning

confidence: 99%

On the Pointwise Bishop–Phelps–Bollobás Property for Operators

Dantas¹,

Kadets²,

Kim³

et al. 2018

Can. J. Math.-J. Can. Math.

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We study approximation of operators between Banach spaces X and Y that nearly attain their norms in a given point by operators that attain their norms at the same point. When such approximations exist, we say that the pair (X, Y ) has the pointwise Bishop-Phelps-Bollobás property (pointwise BPB property for short). In this paper we mostly concentrate on those X, called universal pointwise BPB domain spaces, such that (X, Y ) possesses pointwise BPB property for every Y , and on those Y , called universal pointwise BPB range spaces, such that (X, Y ) enjoys pointwise BPB property for every uniformly smooth X. We show that every universal pointwise BPB domain space is uniformly convex and that Lp(µ) spaces fail to have this property when p > 2. For universal pointwise BPB range space, we show that every simultaneously uniformly convex and uniformly smooth Banach space fails it if its dimension is greater than one. We also discuss a version of the pointwise BPB property for compact operators.arXiv:1709.00032v2 [math.FA] 26 Sep 2018 1 , Y ) fails the BPBp (even though all elements in L( 2 1 , Y ) attain their norms) [3]. Moreover, such Y can be found in a way that for every Banach space X, the set of all norm attaining operators is dense in L(X, Y ) [8, Theorem 4.2]. For more results and background on the BPBp, we refer the reader to the recent papers [1,6,9,10] and the references therein.In the very recent paper [13], the following stronger version of the Bishop-Phelps-Bollobás property was introduced (with the name of Bishop-Phelps-Bollobás point property).Definition 1.1 ([13, Definition 1.2]). We say that a pair (X, Y ) of Banach spaces has the pointwise Bishop-Phelps-Bollobás property (pointwise BPB property, for short) if given ε > 0, there existsη(ε) > 0 such that whenever T ∈ L(X, Y ) with T = 1 and x 0 ∈ S X satisfyThe difference between the BPBp and this new property is that, in the last one, the point x 0 does not move and the new operator S attains its norm at it. Also observe that, by an easy change of parameters, we may consider T ∈ L(X, Y ) with T 1 in the above definition, and we will use this fact without any explicit reference. Nevertheless, the same trick does not work for the point x 0 (as the operator S attains its norm at x 0 ), so we have to consider x 0 ∈ S X .Another useful definitions, motivated by the corresponding ones for the BPBp given in [8], are the following.Definition 1.2. A Banach space X is said to be a universal pointwise BPB domain space if (X, Y ) has the pointwise BPB property for every Banach space Y . A Banach space Y is said to be a universal pointwise BPB range space if (X, Y ) has the pointwise BPB property for every uniformly smooth Banach space X.As we already mentioned, the pointwise BPB property was introduced in [13] where, among other results, it was proved that if (X, Y ) has the pointwise BPB property then X must be uniformly smooth, that Hilbert spaces are universal pointwise BPB domain spaces, and that the class of universal pointwise BPB range spaces includes uni...

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“…There have been many efforts handling the heredity of norm attainment for operators and Lipschitz maps. We refer to [3,6] for these kinds of study which have been done recently, and we basically follow their ideas. Recall that an absolute norm |…”

Section: Denseness Of Norm Attaining Lipschitz Maps Toward Vectorsmentioning

confidence: 99%

Norm attaining Lipschitz maps toward vectors

Choi¹

2022

Preprint

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We extend the recent result of G. Godefroy which concerns the existence of non-norm attaining Lipschitz maps in order to characterize the norm attainment toward vectors for Lipschitz maps in the general setting of underlying space. The main theorem of the present paper states that the existence of nonnorm attaining Lipschitz maps toward vectors on a large class of metric spaces is characterized by the finite-dimensionality of range space. As an extension of his counterexample, some denseness results on norm attaining Lipschitz maps toward vectors are also shown.Definition 1.1. A Lipschitz map f ∈ Lip 0 (M, Y ) is said to attain its norm toward y ∈ Y if there exists a sequence of distinct pairs {(p n , q n )} ⊆ M × M such that

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The Bishop–Phelps–Bollobás Property and Absolute Sums

Cited by 8 publications

References 29 publications

The Bishop–Phelps–Bollobás property for Lipschitz maps

The Bishop–Phelps–Bollobás property for Lipschitz maps

On the Pointwise Bishop–Phelps–Bollobás Property for Operators

Norm attaining Lipschitz maps toward vectors

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