Numerical radius attaining compact linear operators

Capel, Ángela; Martı́n, Miguel; Merí, Javier

doi:10.1016/j.jmaa.2016.02.074

Cited by 7 publications

(6 citation statements)

References 25 publications

(38 reference statements)

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“…As both C[0, 1] and L 1 [0, 1] have the BPBp for numerical radius [6,23], this shows that there is no possible valid reciprocal results for Theorem 4.1 and Proposition 4.7. On the other hand, the sets NRA(C[0, 1] ⊕ 1 L 1 [0, 1]) ∩ K(C[0, 1] ⊕ 1 L 1 [0, 1]) and NRA(C[0, 1] ⊕ ∞ L 1 [0, 1]) ∩ K(C[0, 1] ⊕ ∞ L 1 [0, 1]) are dense in K(C[0, 1] ⊕ 1 L 1 [0, 1]) and K(C[0, 1] ⊕ ∞ L 1 [0, 1]), respectively (see [12,Example 3.4]). Nevertheless, we do not know if there is some reciprocal result for the BPBp-nu for compact operators.…”

Section: Results For Numerical Radiusmentioning

confidence: 99%

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The Bishop–Phelps–Bollobás Property and Absolute Sums

et al. 2019

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In this paper we study conditions assuring that the Bishop-Phelps-Bollobás property (BPBp, for short) is inherited by absolute summands of the range space or of the domain space. Concretely, given a pair (X, Y ) of Banach spaces having the BPBp, (a) if Y 1 is an absolute summand of Y , then (X, Y 1 ) has the BPBp; (b) if X 1 is an absolute summand of X of type 1 or ∞, then (X 1 , Y ) has the BPBp. Besides, analogous results for the BPBp for compact operators and for the density of norm attaining operators are also given. We also show that the Bishop-Phelps-Bollobás property for numerical radius is inherited by absolute summands of type 1 or ∞. Moreover, we provide analogous results for numerical radius attaining operators and for the BPBp for numerical radius for compact operators.This shows that |w * 1 (Sw 1 )| = v( S) v(S). So, |w * 1 (Sw 1 )| = v(S) and W has the weak BPBp-nu.Again, the above proof can be adapted to compact operators to get the following result.Proposition 4.5. Let X be a Banach space and let W be an absolute summand of type 1 or ∞ of X. If X has the weak BPBp-nu for compact operators, so does W .

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Section: Results For Numerical Radiusmentioning

confidence: 99%

“…We denote by NRA(X) the set of all numerical radius attaining operators on X. We refer the reader to the classical books [9,10] for background on numerical radius of operators and to [1,12,32] and the references therein for background on the study of the density of the set of numerical radius attaining operators.…”

Section: Introductionmentioning

confidence: 99%

The Bishop–Phelps–Bollobás Property and Absolute Sums

et al. 2019

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“…Stability on the denseness. Started from [20], there have been many efforts for instance in [3,9,15] to construct an operator which cannot be approximated by numerical radius attaining operators. However, all of those counterexamples are based on a non-injective operator, so we cannot apply the same kind of argument to the case of the Crawford number attaining operators.…”

Section: Questions and Remaining Problemsmentioning

confidence: 99%

“…The study of numerical radius was one of main interest in the field of functional analysis in recent decades. In particular, phenomenon of numerical radius attaining operators was discovered in many sources such as [1,2,3,4,9,20]. It has not been that long since the minimum norm of an operator became an issue as a separated topic.…”

Section: Introductionmentioning

confidence: 99%

On the Crawford number attaining operators

Choi¹,

Lee²

2022

Preprint

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We study the denseness of Crawford number attaining operators on Banach spaces. Mainly, we prove that if a Banach space has the RNP, then the set of Crawford number attaining operators is dense in the space of bounded linear operators. We also see among others that the set of Crawford number attaining operators may be dense in the space of all bounded linear operators while they do not coincide, by observing the case of compact operators when the Banach space has a 1-unconditional basis. Furthermore, we show a Bishop-Phelps-Bollobás type property for the Crawford number for certain Banach spaces, and we finally discuss some difficulties and possible problems on the topic.

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“…Recall that for a Banach space X, a bounded linear operator T ∈ L(X, X) is said to be a numerical radius attaining operator when the supremum in the equation ( 1) is attained by an element (x, x * ) such that x * (x) = 1. Numerical radius attaining operators, introduced in [40], have been investigated by many researchers (see, for instance, [1,8,9,37] and the references therein). In a similar spirit, we now define a Lipschitz numerical radius attaining map.…”

Section: Lipschitz Numerical Radii Indices and Daugavet Centersmentioning

confidence: 99%

On the Lipschitz numerical index of Banach spaces

Choi¹,

Jung²,

Tag³

2021

Preprint

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We provide some new consequences on the Lipschitz numerical radius and index which were introduced recently. More precisely, we give some renorming results on the Lipschitz numerical index, introduce a concept of Lipschitz numerical radius attaining functions in order to show that the denseness fails for an arbitrary Banach space, and study a Lipschitz version of Daugavet centers. Furthermore, we discuss the Lipschitz numerical index of vector-valued function spaces, absolute sums of Banach spaces, the Köthe-Bochner spaces, and Banach spaces which contain a dense union of increasing family of one-complemented subspaces.

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Numerical radius attaining compact linear operators

Abstract: Abstract. We show that there are compact linear operators on Banach spaces which cannot be approximated by numerical radius attaining operators.

Cited by 7 publications

References 25 publications

The Bishop–Phelps–Bollobás Property and Absolute Sums

The Bishop–Phelps–Bollobás Property and Absolute Sums

On the Crawford number attaining operators

On the Lipschitz numerical index of Banach spaces

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