Finite-dimensional Banach spaces with numerical index zero

Martı́n, Miguel; Merí, Javier; Rodríguez-Palacios, A.

doi:10.1512/iumj.2004.53.2447

Cited by 15 publications

(16 citation statements)

References 18 publications

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“…It is a well known result (see [14,Corollary 2.5] and [15, Theorem 3.1]) that the only two dimensional real space with numerical index 0 is the Euclidean space. The above theorem allows us to give a different and elementary proof of this fact.…”

Section: The Resultsmentioning

confidence: 99%

Two-dimensional Banach spaces with polynomial numerical index zero

Garcı́a

Grecu

Maestre

et al. 2009

Linear Algebra and its Applications

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We study two-dimensional Banach spaces with polynomial numerical indices equal to zero.Recently, Y. S. Choi, D. García, S. G. Kim and M. Maestre [2] have introduced the polynomial numerical index of order k of a Banach space X as the constant n (k) (X) defined by n (k) (X) = max c 0 : c P v(P ) ∀ P ∈ P k X; X = inf v(P ) : P ∈ P k X; X , P = 1 for every k ∈ N. This concept is a generalization of the numerical index of a Banach space (recovered for k = 1) which was first suggested by G. Lumer in 1968 [7].Let us recall some facts about the polynomial numerical index which are relevant to our discussion. We refer the reader to the already cited [2] and to [4,12,13] for recent results and background. The easiest examples are n (k) (R) = 1 and n (k) (C) = 1 for every k ∈ N. In the complex case, n (k) (C(K)) = 1 for every k ∈ N and n (2) (ℓ 1 ) 1 2 . The real spaces ℓ m 1 , ℓ m ∞ , c 0 , ℓ 1 and ℓ ∞ have polynomial numerical index of order 2 equal to 1/2 ([12]). The only finite-dimensional real Banach space X with n (2) (X) = 1 is X = R ([13]). The inequality n (k+1) (X) n (k) (X) holds for 2000 Mathematics Subject Classification. Primary 46B04; Secondary 46B20, 46G25, 47A12.

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Section: The Resultsmentioning

confidence: 99%

Two-dimensional Banach spaces with polynomial numerical index zero

Garcı́a

Grecu

Maestre

et al. 2009

Linear Algebra and its Applications

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“…For instance, if ⊕ a = ⊕ 2 , then n (X R ) = 1, while if ⊕ a = ⊕ ∞ or ⊕ a = ⊕ 1 , then 1/2 n (X R ) √ 3/2 (see corollary 4.8). Finite-dimensional real spaces with numerical index zero were characterized in [24], where a structure result is given. By proposition 2.1, the second numerical index of all of them is positive.…”

Section: Some Open Problemsmentioning

confidence: 99%

On a second numerical index for Banach spaces

Kim¹,

Lee²,

Martı́n³

et al. 2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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We introduce a second numerical index for real Banach spaces with non-trivial Lie algebra, as the best constant of equivalence between the numerical radius and the quotient of the operator norm modulo the Lie algebra. We present a number of examples and results concerning absolute sums, duality, vector-valued function spaces…which show that, in many cases, the behaviour of this second numerical index differs from the one of the classical numerical index. As main results, we prove that Hilbert spaces have second numerical index one and that they are the only spaces with this property among the class of Banach spaces with one-unconditional basis and non-trivial Lie algebra. Besides, an application to the Bishop-Phelps-Bollobás property for the numerical radius is given.

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“…Therefore, as we may nd polyhedral norms as close to the norm of Y as we want, there exists a polyhedral norm such that, calling W to the space Y endowed with the this norm, we still have n comp (W ) < n r (W ). Moreover, since W is polyhedral it cannot contain an isometric copy of C, so Theorem 2.4 in [15] tells us that n comp (W ) = 0. Now, we may follow the lines of the proof of Example 4.1 and consider the space…”

Section: Some Examples and Remarksmentioning

confidence: 99%