The Numerical Index of a Normed Space

Duncan, John F. R.; McGregor, Colin; Pryce, John D.; White, A. J.

doi:10.1112/jlms/2.part_3.481

Cited by 90 publications

(76 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The question whether this is actually an equality seems to be open. Second, it is a classical result that L-and M -spaces have numerical index 1 [6].…”

Section: N(x) = Max{k ≥ 0 : K T ≤ V(t ) ∀ T ∈ L(x)}mentioning

confidence: 99%

Banach spaces having the Radon-Nikodym property and numerical index 1

Martı́n

2003

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…The question whether this is actually an equality seems to be open. Second, it is a classical result that L-and M -spaces have numerical index 1 [6].…”

Section: N(x) = Max{k ≥ 0 : K T ≤ V(t ) ∀ T ∈ L(x)}mentioning

confidence: 99%

Banach spaces having the Radon-Nikodym property and numerical index 1

Martı́n

2003

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…Indeed, it is clear that x k ∈ S(B X , x * , ε/n), which ensures validity of (4). The remaining condition (5) follows from (6) and (7):…”

Section: A Reformulation Of Lushnessmentioning

confidence: 99%

“…The numerical index of a Banach space X was introduced by G. Lumer in 1968 (see [6]) and it is the best constant of equivalence between the numerical radius and the usual norm of operators on L(X). Concretely, n(X) = inf{v(T ) : T ∈ L(X), T = 1} = max{k 0 : k T v(T ) ∀T ∈ L(X)}.…”

Section: Introductionmentioning

confidence: 99%

Properties of lush spaces and applications to Banach spaces with numerical index 1

et al. 2009

View full text Add to dashboard Cite

Abstract. The concept of lushness was introduced recently as a Banach space property, which ensures that the space has numerical index 1. We prove that for Asplund spaces lushness is actually equivalent to numerical index 1. We prove that every separable Banach space containing an isomorphic copy of c 0 can be renormed equivalently to be lush, and thus to have numerical index 1. The rest of the paper is devoted to the study of lushness just as a property of Banach spaces. We prove that lushness is separably determined, is stable under ultraproducts, and we characterize those spaces of the form X = (R n , · ) with absolute norm such that X-sum preserves lushness of summands, showing that this property is equivalent to lushness of X.

show abstract

“…Several years later, Duncan et al . [11] proved that L-spaces and M -spaces have numerical index 1. McGregor [18] obtained necessary and sufficient conditions such that a finite-dimensional normed space has numerical index 1.…”

Section: A Mapping P : E → F Is Called a Continuous K-homogeneous Polmentioning

confidence: 99%

The Polynomial Numerical Index of a Banach Space

Choi¹,

Garcı́a²,

Kim³

et al. 2006

Proceedings of the Edinburgh Mathematical Society

View full text Add to dashboard Cite

In this paper, we introduce the polynomial numerical index of order k of a Banach space, generalizing to k-homogeneous polynomials the 'classical' numerical index defined by Lumer in the 1970s for linear operators. We also prove some results. Let k be a positive integer. We then have the following:is sharp.(iii) The inequalities(iv) The relation between the polynomial numerical index of c 0 , l 1 , l∞ sums of Banach spaces and the infimum of the polynomial numerical indices of them.(v) The relation between the polynomial numerical index of the space C(K, E) and the polynomial numerical index of E.(vi) The inequality n (k) (E * * ) n (k) (E) for every Banach space E.Finally, some results about the numerical radius of multilinear maps and homogeneous polynomials on C(K) and the disc algebra are given.

show abstract

The Numerical Index of a Normed Space

Cited by 90 publications

References 3 publications

Banach spaces having the Radon-Nikodym property and numerical index 1

Banach spaces having the Radon-Nikodym property and numerical index 1

Properties of lush spaces and applications to Banach spaces with numerical index 1

The Polynomial Numerical Index of a Banach Space

Contact Info

Product

Resources

About