The Bishop–Phelps–Bollobás theorem for operators

Acosta, Marı́a D.; Aron, Richard M.; Garcı́a, Domingo; Maestre, Manuel

doi:10.1016/j.jfa.2008.02.014

Cited by 113 publications

(225 citation statements)

References 9 publications

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“…Bishop-Phelps-Bollobás theorem does not holds for L( (2) 1 , Y ) [4,10] ( (2) 1 is the two-dimensional L 1 space) while all elements of L( (2) 1 , Y ) attain their norms. We refer the reader to [2,4,5,10,11,12,17,21,22,29,30,31] and references therein for more information and background.…”

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confidence: 99%

“…We refer the reader to [2,4,5,10,11,12,17,21,22,29,30,31] and references therein for more information and background. Let us just mention some examples of pairs of classical spaces having the Bishop-Phelps-Bollobás property for operators, namely, (L 1 (µ), L ∞ (ν)) for arbitrary measure µ and localizable measure ν [12,22], (L 1 (µ), L 1 (ν)) for arbitrary measures µ and ν [22], (C(K 1 ), C(K 2 )) for every compact spaces K 1 and K 2 [5], and (L p (µ), Y ) for arbitrary measure µ, arbitrary Banach space Y and 1 < p < ∞ [6,30].…”

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confidence: 99%

“…None of the reversed results are true: the pair ( 1 , 1 ) fails the Bishop-Phelps-Bollobás for bilinear forms [23], while norm attaining bilinear forms are dense in B( 1 × 1 ) [13] and also the pair ( 1 , ∞ ) has the Bishop-Phelps-Bollobás property for operators [4].…”

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confidence: 99%

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Bishop–Phelps–Bollobás property for bilinear forms on spaces of continuous functions

2015

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Abstract. It is shown that the Bishop-Phelps-Bollobás theorem holds for bilinear forms on the complex C 0 (L 1 ) × C 0 (L 2 ) for arbitrary locally compact topological Hausdorff spaces L 1 and L 2 .All along the paper, we will use the following usual notation. Let B X and S X denote, respectively, the closed unit ball and the unit sphere of a Banach space X, and let X * denotes the (topological) dual space of X. We write L(X, Y ) to denote the space of all bounded linear operators from a Banach space X into a Banach space Y and B(X × Y ) to denote the space of all bounded bilinear forms defined on X × Y . We say that T ∈ L(X, Y ) (respectively B ∈ B(X × Y )) attains its norm if there is x ∈ S X such that T x = T (respectively, there are x ∈ S X and y ∈ S Y such that |B(x, y)| = B ).The classical Bishop-Phelps theorem [14] states that the set of norm attaining linear functionals are dense in the topological dual of an arbitrary Banach space. Bollobás Let X be a Banach space. If x ∈ B X and x * ∈ S B * X satisfy Re x * (x) > 1 − ε 2 /2. Then there exist y ∈ S X and y * ∈ S X * such that y * (y) = 1, x * − y * < ε and x − y < ε. there exist x 1 ∈ S X and S ∈ L(X, Y ) such that S(x 1 ) = S = 1, x 0 − x 1 < ε and S − B < ε. In this case, it also said that the Bishop-Phelps-Bollobás theorem holds for L(X, Y ).It is clear that the Bishop-Phelps-Bollobás property for a pair (X, Y ) implies that norm attaining operators are dense in L(X, Y ), being false the converse: there is a (reflexive) space Y such that the

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Bishop–Phelps–Bollobás property for bilinear forms on spaces of continuous functions

2015

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“…Aron, García and Maestre in [3] where the following result is proved: the Bishop-PhelpsBollobás theorem holds for L(ℓ 1 ; Y ) if and only if Y has the so called approximate hyperplane series property. Also, the authors study a quantitative version of the Lindenstrauss theorem for operators, which will be referred to as a Lindenstrauss-Bollobás-type theorem.…”

Section: On Quantitative Versions Of Bishop-phelps and Lindenstrauss mentioning

confidence: 99%

A Lindenstrauss theorem for some classes of multilinear mappings

Carando

Lassalle

Mazzitelli

2015

Journal of Mathematical Analysis and Applications

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Abstract. Under some natural hypotheses, we show that if a multilinear mapping belongs to some Banach multlinear ideal, then it can be approximated by multilinear mappings belonging to the same ideal all whose Arens extensions attain their norms at the same point. We prove a similar result for the class of symmetric multilinear mappings. We see that the quantitative (Bollobás-type) version of these results fails in every multilinear ideal.

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“…In 2008, Acosta, Aron, García and Maestre in [1] introduced the following Bishop-PhelpsBollobás property as an extension of the Bishop-Phelps-Bollobás theorem to the vectorvalued case.…”

Section: Proposition 1 ([5 Corollary 24])mentioning

confidence: 99%

Quantitative version of the Bishop-Phelps-Bollobas theorem for operators with values in a space with the property $\beta$

Kadets

Soloviova

2017

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The Bishop-Phelps-Bollobás property for operators deals with simultaneous approximation of an operator T and a vector x at which T : X → Y nearly attains its norm by an operator F and a vector z, respectively, such that F attains its norm at z. We study the possible estimates from above and from below for parameters that measure the rate of approximation in the Bishop-Phelps-Bollobás property for operators for the case of Y having the property β of Lindenstrauss.

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The Bishop–Phelps–Bollobás theorem for operators

Cited by 113 publications

References 9 publications

Bishop–Phelps–Bollobás property for bilinear forms on spaces of continuous functions

Bishop–Phelps–Bollobás property for bilinear forms on spaces of continuous functions

A Lindenstrauss theorem for some classes of multilinear mappings

Quantitative version of the Bishop-Phelps-Bollobas theorem for operators with values in a space with the property $\beta$

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