On the numerical index of absolute symmetric norms on the plane

Merí, Javier; Quero, Alicia

doi:10.1080/03081087.2020.1762532

Cited by 7 publications

(3 citation statements)

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“…All these results can be found in the cited papers [10,12,14]. Some recent results can be found in [18], where the exact value of some two-dimensional p spaces is calculated, and in [1,2,22], for instance. Different extensions of the concept of numerical index appear in [11] and [25].…”

Section: Introductionmentioning

confidence: 99%

Numerical index and Daugavet property of operator ideals and tensor products

Martín¹,

Merí²,

Quero³

2020

Preprint

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We show that the numerical index of any operator ideal is less than or equal to the minimum of the numerical indices of the domain and the range. Further, we show that the numerical index of the ideal of compact operators or the ideal of weakly compact operators is less than or equal to the numerical index of the dual of the domain, and this result provides interesting examples. We also show that the numerical index of a projective or injective tensor product of Banach spaces is less than or equal to the numerical index of any of the factors. Finally, we show that if a projective tensor product of two Banach spaces has the Daugavet property and the unit ball of one of the factor is slicely countably determined or its dual contains a point of Fréchet differentiability of the norm, then the other factor inherits the Daugavet property. If an injective tensor product of two Banach spaces has the Daugavet property and one of the factors contains a point of Fréchet differentiability of the norm, then the other factor has the Daugavet property.

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Section: Introductionmentioning

confidence: 99%

Numerical index and Daugavet property of operator ideals and tensor products

Martín¹,

Merí²,

Quero³

2020

Preprint

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“…The numerical index of the projective and the injective tensor product of two Banach spaces is less than or equal to the minimum of the numerical indexes of both factors [61]. Many other recent publications have been devoted to the study of the numerical index (see, for example, [20,21,52,60,65], there are over 1600 papers published under the MSC item 47A12 "Numerical range, numerical radius").…”

Section: Introductionmentioning

confidence: 99%

Estimations of the numerical index of a JB$^*$-triple

Cabezas¹,

Peralta²

2023

Preprint

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We prove that every commutative JB * -triple has numerical index one. We also revisit the notion of commutativity in JB * -triples to show that a JBW * -triple M has numerical index one precisely when it is commutative, while e −1 ≤ n(M ) ≤ 2 −1 otherwise. Consequently, a JB * -triple E is commutative if and only if n(E * ) = 1 (equivalently, n(E * * ) = 1). In the general setting we prove that the numerical index of each JB * -triple E admitting a non-commutative element also satisfies e −1 ≤ n(M ) ≤ 2 −1 , and the same holds when the bidual of E contains a Cartan factor of rank ≥ 2 in its atomic part.

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“…Very recently, it has been proved [11] that the numerical index is attained at the operator 0 1 −1 0 for many absolute and symmetric norms on R 2 and, as a major consequence, it is shown that n(ℓ 2 p ) = M p for 3 2 p 3. In [12] the authors polished skilfully the arguments of [11] to provide a slight improvement: the equality n(ℓ 2 p ) = M p is proved for 1 + α 0 p α 1 where α 0 is the root of f…”

Section: Introductionmentioning

confidence: 99%