Nonlinear Geometry

Guirao, Antonio José; Montesinos, Vicente; Zizler, Václav

doi:10.1007/978-3-319-33572-8_5

Cited by 2 publications

(2 citation statements)

References 40 publications

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“…Ostrovskii asked in [24, §12, pg. 65] whether there exist infinite-dimensional Banach spaces on which every operator attains its norm (this question is also asked in [20, Problem 8] and [15, Problem 217]). By Holub’s theorem [16], if such an infinite-dimensional Banach space exists, it cannot have the AP.…”

Section: The Resultsmentioning

confidence: 99%

On the Existence of Non-Norm-Attaining Operators

Dantas¹,

Jung

Martínez-Cervantes³

2021

J. Inst. Math. Jussieu

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In this article, we provide necessary and sufficient conditions for the existence of non-norm-attaining operators in $\mathcal {L}(E, F)$ . By using a theorem due to Pfitzner on James boundaries, we show that if there exists a relatively compact set K of $\mathcal {L}(E, F)$ (in the weak operator topology) such that $0$ is an element of its closure (in the weak operator topology) but it is not in its norm-closed convex hull, then we can guarantee the existence of an operator that does not attain its norm. This allows us to provide the following generalisation of results due to Holub and Mujica. If E is a reflexive space, F is an arbitrary Banach space and the pair $(E, F)$ has the (pointwise-)bounded compact approximation property, then the following are equivalent: (i) $\mathcal {K}(E, F) = \mathcal {L}(E, F)$ ; (ii) Every operator from E into F attains its norm; (iii) $(\mathcal {L}(E,F), \tau _c)^* = (\mathcal {L}(E, F), \left \Vert \cdot \right \Vert )^*$ , where $\tau _c$ denotes the topology of compact convergence. We conclude the article by presenting a characterisation of the Schur property in terms of norm-attaining operators.

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Section: The Resultsmentioning

confidence: 99%

On the Existence of Non-Norm-Attaining Operators

Dantas¹,

Jung

Martínez-Cervantes³

2021

J. Inst. Math. Jussieu

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“…They are renormings of , and moreover they do not admit even any infinite equilateral set (However, all equivalent renormings of for uncountable admit -separated set of size by Remark 3.16 of [19]). The above theorem solves Problem 293 of [16].…”

Section: Introductionmentioning

confidence: 99%

Banach Spaces in Which Large Subsets of Spheres Concentrate

Koszmider

2022

J. Inst. Math. Jussieu

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We construct a nonseparable Banach space $\mathcal {X}$ (actually, of density continuum) such that any uncountable subset $\mathcal {Y}$ of the unit sphere of $\mathcal {X}$ contains uncountably many points distant by less than $1$ (in fact, by less then $1-\varepsilon $ for some $\varepsilon>0$ ). This solves in the negative the central problem of the search for a nonseparable version of Kottman’s theorem which so far has produced many deep positive results for special classes of Banach spaces and has related the global properties of the spaces to the distances between points of uncountable subsets of the unit sphere. The property of our space is strong enough to imply that it contains neither an uncountable Auerbach system nor an uncountable equilateral set. The space is a strictly convex renorming of the Johnson–Lindenstrauss space induced by an $\mathbb {R}$ -embeddable almost disjoint family of subsets of $\mathbb {N}$ . We also show that this special feature of the almost disjoint family is essential to obtain the above properties.

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Nonlinear Geometry

Cited by 2 publications

References 40 publications

On the Existence of Non-Norm-Attaining Operators

On the Existence of Non-Norm-Attaining Operators

Banach Spaces in Which Large Subsets of Spheres Concentrate

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