A counterexample on numberical radius attaining operators

Payá, Rafael

doi:10.1007/bf02764803

Cited by 27 publications

(30 citation statements)

References 13 publications

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“…On the other hand, some attention has been paid to the study of denseness of numerical radius attaining mappings (see, for instance, [1,4,5,13,16,18,24]) as a parallel question for the norm. We now look more closely at such mappings and investigate whether they attain their norms or numerical radii at extreme points.…”

Section: Introductionmentioning

confidence: 99%

Norm or numerical radius attaining polynomials on C(K)

Choi

Garcı́a

Kim

et al. 2004

Journal of Mathematical Analysis and Applications

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Let C(K, C) be the Banach space of all complex-valued continuous functions on a compact Hausdorff space K. We study when the following statement holds: every norm attaining n-homogeneous complex polynomial on C(K, C) attains its norm at extreme points. We prove that this property is true whenever K is a compact Hausdorff space of dimension less than or equal to one. In the case of a compact metric space a characterization is obtained. As a consequence we show that, for a scattered compact Hausdorff space K, every continuous n-homogeneous complex polynomial on C(K, C) can be approximated by norm attaining ones at extreme points and also that the set of all extreme points of the unit ball of C(K, C) is a norming set for every continuous complex polynomial. Similar results can be obtained if "norm" is replaced by "numerical radius."

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Section: Introductionmentioning

confidence: 99%

Norm or numerical radius attaining polynomials on C(K)

Choi

Garcı́a

Kim

et al. 2004

Journal of Mathematical Analysis and Applications

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“…Among the positive results on this topic, we would like to mention that the set of numerical radius attaining operators is dense for Banach spaces with the Radon-Nikodým property (M. Acosta and R. Payá [5]) and for L 1 (µ) spaces (M. Acosta [1]) and C(K) spaces (C. Cardassi [11]). On the other hand, the first example of Banach space for which the set of numerical radius attaining operators is not dense was given by R. Payá in 1992 [26]. Another counterexample was discovered shortly later by M. Acosta, F. Aguirre and R. Payá [4].…”

Section: It Is Clear That V Is a Continuous Seminorm Which Satisfies mentioning

confidence: 99%

“…We will make use of the following lemma from [26], which we state for the convenience of the reader. …”

Section: The Examplementioning

confidence: 99%

Numerical radius attaining compact linear operators

Capel

Martı́n

Merí

2017

Journal of Mathematical Analysis and Applications

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“…Extensive work has been developed over recent years for classical numerical radius attaining operators (see [1,4,12,16] and the references therein). In Theorem 3.2 we generalize, for the context of convex numerical radius, the following version of James's theorem for numerical radius [2, Theorem 1]: if every rank-one operator on E attains its numerical radius, then E is reflexive.…”

Section: Operators That Attain the Convex Numerical Radiusmentioning

confidence: 99%