Octahedral norms in spaces of operators

Guerrero, Julio Becerra; López-Pérez, Ginés; Zoca, Abraham Rueda

doi:10.1016/j.jmaa.2015.02.046

Cited by 19 publications

(33 citation statements)

References 10 publications

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“…Note that Theorem (1) with

δ = 2

is exactly Theorem . Combining Theorems and improves [, Corollary 3.11] as follows. Corollary Let X and Y be Banach spaces, and let H be a closed subspace of

L (X, Y)

containing the finite rank operators.…”

Section: Necessary Conditions For Roughness In Spaces Of Operatorsmentioning

confidence: 90%

“…In particular, it shows that

L (c_{0}, c_{0})

is octahedral, while its octahedrality can not be deduced from Theorems or . Also, Theorem allows one to refine [, Corollaries 3.3, 3.4, and 3.6].…”

Section: Sufficient Conditions For Roughness In Spaces Of Operatorsmentioning

confidence: 99%

“…We do not know whether for the octahedrality of

L (X, Y)

it is, in general, sufficient that only one of the spaces

X^{*}

or Y is octahedral without any additional assumptions (see also the discussion after Corollary 3.6 in ). We next show that, for

1 < p < \infty

, the space

L (c_{0}, ℓ_{p}^{2})

is octahedral.…”

Section: Sufficient Conditions For Roughness In Spaces Of Operatorsmentioning

confidence: 99%

“…In Section , we first establish a quantitative version of the following Theorem from in terms of average roughness (see Theorem ). Theorem Let X and Y be Banach spaces, and let H be a closed subspace of

L (X, Y)

containing the finite rank operators.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Rough norms in spaces of operators

Haller

Langemets

Põldvere

2017

Mathematische Nachrichten

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We investigate sufficient and necessary conditions for the space of bounded linear operators between two Banach spaces to be rough or average rough. Our main result is that L(X,Y) is δ‐average rough whenever X* is δ‐average rough and Y is alternatively octahedral. This allows us to give a unified improvement of two theorems by Becerra Guerrero, López‐Pérez, and Rueda Zoca [J. Math. Anal. Appl. 427 (2015)].

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“…Note that Theorem (1) with

δ = 2

is exactly Theorem . Combining Theorems and improves [, Corollary 3.11] as follows. Corollary Let X and Y be Banach spaces, and let H be a closed subspace of

L (X, Y)

containing the finite rank operators.…”

Section: Necessary Conditions For Roughness In Spaces Of Operatorsmentioning

confidence: 90%

“…In particular, it shows that

L (c_{0}, c_{0})

is octahedral, while its octahedrality can not be deduced from Theorems or . Also, Theorem allows one to refine [, Corollaries 3.3, 3.4, and 3.6].…”

Section: Sufficient Conditions For Roughness In Spaces Of Operatorsmentioning

confidence: 99%

“…We do not know whether for the octahedrality of

L (X, Y)

it is, in general, sufficient that only one of the spaces

X^{*}

or Y is octahedral without any additional assumptions (see also the discussion after Corollary 3.6 in ). We next show that, for

1 < p < \infty

, the space

L (c_{0}, ℓ_{p}^{2})

is octahedral.…”

Section: Sufficient Conditions For Roughness In Spaces Of Operatorsmentioning

confidence: 99%

L (X, Y)

containing the finite rank operators.…”

Section: Introductionmentioning

confidence: 99%

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Rough norms in spaces of operators

Haller

Langemets

Põldvere

2017

Mathematische Nachrichten

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“…In recent years, these properties have been intensively studied (see, e.g., [1][2][3][4][5][6][7][8][9][10][11] for some typical results and further references).…”

mentioning

confidence: 99%

Diametral strong diameter two property of Banach spaces is stable under direct sums with 1-norm

Haller

Pirk

Põldvere

2016

ACUTM

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Abstract. We prove that the diametral strong diameter 2 property of a Banach space (meaning that, in convex combinations of relatively weakly open subsets of its unit ball, every point has an "almost diametral" point) is stable under 1-sums, i.e., the direct sum of two spaces with the diametral strong diameter 2 property equipped with the 1-norm has again this property.All Banach spaces considered in this note are over the real field. The closed unit ball and the unit sphere of a Banach space X will be denoted by B X and S X , respectively. Whenever referring to a relative weak topology, we mean such a topology on the closed unit ball of the space under consideration.Diameter 2 properties for a Banach space mean that certain subsets of its unit ball (e.g., slices, nonempty relatively weakly open subsets, or convex combinations of weakly open subsets) have diameter equal to 2. In recent years, these properties have been intensively studied (see, e.g., [1][2][3][4][5][6][7][8][9][10][11] for some typical results and further references).To clarify the gap between the well-studied Daugavet property [12] and known diameter 2 properties, the diametral diameter 2 properties were introduced and studied in the recent preprint [7]. In particular, the stability under p-sums of diametral diameter 2 properties was analyzed. The question whether the 1-sum of two Banach spaces enjoying the diametral strong diameter 2 property also has this property, was posed as an open problem in [7]. Below, we shall answer this question in the affirmative.

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Strong Diameter Two Property and Convex Combinations of Slices Reaching the Unit Sphere

2019

Self Cite

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We characterise the class of those Banach spaces in which every convex combination of slices of the unit ball intersects the unit sphere as the class of those spaces in which every convex combination of slices of the unit ball contains two points at distance exactly two. Also, we study when the convex combinations of slices of the unit ball are relatively open or has non-empty relative interior for different topologies, studying the relationship between them and studying these properties for L∞-spaces and preduals of L 1 -spaces.

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Octahedral norms in spaces of operators

Cited by 19 publications

References 10 publications

Rough norms in spaces of operators

Rough norms in spaces of operators

Diametral strong diameter two property of Banach spaces is stable under direct sums with 1-norm

Strong Diameter Two Property and Convex Combinations of Slices Reaching the Unit Sphere

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