On the numerical index with respect to an operator

Kadets, Vladimir; Martı́n, Miguel; Merí, Javier; Pérez, Aquilino Sánchez; Quero, Alicia

doi:10.4064/dm805-9-2019

Cited by 6 publications

(5 citation statements)

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“…The space of bounded linear operators on X endowed with the numerical radius norm is denoted by (L(X)) w . For a detailed study on numerical range of operators and their possible applications, we refer the readers to [4,5,11,13,14]. Birkhoff-James orthogonality is an important tool in the study of smoothness of elements in a given Banach space.…”

Section: Introductionmentioning

confidence: 99%

Numerical radius and a notion of smoothness in the space of bounded linear operators

Roy

Sain

2021

Bulletin des Sciences Mathématiques

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We observe that the classical notion of numerical radius gives rise to a notion of smoothness in the space of bounded linear operators on certain Banach spaces, whenever the numerical radius is a norm. We characterize Birkhoff-James orthogonality in the space of bounded linear operators on a finite-dimensional Banach space, endowed with the numerical radius norm. Some examples are also discussed to illustrate the geometric differences between the numerical radius norm and the usual operator norm, from the viewpoint of operator smoothness.

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Section: Introductionmentioning

confidence: 99%

Numerical radius and a notion of smoothness in the space of bounded linear operators

Roy

Sain

2021

Bulletin des Sciences Mathématiques

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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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Section: Introductionmentioning

confidence: 99%

Numerical radius and a notion of smoothness in the space of bounded linear operators

Roy¹,

Sain²

2021

Preprint

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We observe that the classical notion of numerical radius gives rise to a notion of smoothness in the space of bounded linear operators on certain Banach spaces, whenever the numerical radius is a norm. We demonstrate an important class of real Banach space X for which the numerical radius defines a norm on L(X), the space of all bounded linear operators on X. We characterize Birkhoff-James orthogonality in the space of bounded linear operators on a finite-dimensional Banach space, endowed with the numerical radius norm. Some examples are also discussed to illustrate the geometric differences between the numerical radius norm and the usual operator norm, from the viewpoint of operator smoothness.

show abstract

On the numerical index with respect to an operator

Cited by 6 publications

References 0 publications

Numerical radius and a notion of smoothness in the space of bounded linear operators

Numerical radius and a notion of smoothness in the space of bounded linear operators

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

Numerical radius and a notion of smoothness in the space of bounded linear operators

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