Finite-Dimensional Normed Linear Spaces with Numerical Index 1

McGregor, Colin

doi:10.1112/jlms/s2-3.4.717

Cited by 16 publications

(19 citation statements)

References 2 publications

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“…We are interested in the study of Banach spaces with numerical index 1. In the finite-dimensional context, a satisfactory characterization of this kind of space was given by C. McGregor [11]. With regard to the infinite-dimensional case, a large family of classical spaces with numerical index 1 was exhibited in the previously cited paper of J. Duncan, C. McGregor, J. Pryce and A.…”

Section: Introductionmentioning

confidence: 99%

Real Banach Spaces with Numerical Index 1

López¹,

Martı́n²,

Payá³

1999

Bulletin of the London Mathematical Society

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Section: Introductionmentioning

confidence: 99%

Real Banach Spaces with Numerical Index 1

López¹,

Martı́n²,

Payá³

1999

Bulletin of the London Mathematical Society

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“…⊕ X n E . First of all, let us remark that if all X k are finitedimensional, by [17,Theorem 3.1] lushness is equivalent to the following property |x * (x)| = 1…”

Section: Theorem 52mentioning

confidence: 99%

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

et al. 2009

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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.

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“…All abstract L and M spaces have numerical index 1 (see [7]). McGregor ( [11]) has characterized finite-dimensional spaces of numerical index 1. The theorem below gives a further example of an index-1 space.…”

Section: = V a {I+£u)vjli+ru)mentioning

confidence: 99%

Mapping Theorems and the Numerical Radius

Crabb

Duncan

McGregor

1972

Proceedings of the London Mathematical Society

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Introduction and notationIn this paper we gather together and examine a variety of problems from the theory of numerical ranges. In § 1 we look at relations between the spectrum and the numerical range, extending results of Rota ([12]) and Hildebrandt ([9]) in Hilbert space theory. For an element a of a complex unital Banach algebra there is the well-known inequality e-1 1|a|| ^v{a) ^ \\a\\;and in § 2 we investigate similar inequalities for real unital Banach algebras; a topic initiated by Bonsall and Duncan ([2]) and developed by Lumer ([10]). The next two sections are concerned with the numerical index. Section 3 deals with relations between the index and the power inequality and also spaces with index 1, while § 4 consists of some examples showing how the index can vary under certain mappings. The numerical range of idempotents is the subject of the final section; the main results being concerned with extremal situations.Given a real or complex Banach space X we shall write X' for the dual space, X x for the closed unit ball of X and S(X) for the set {x e X: \\x\\ = 1}. We shall write B(X) for the algebra of all bounded linear operators on X and |-| for the operator norm on B(X).Let A be a real or complex unital Banach algebra with identity 1. For a E S(A), the set of support functionals for a is defined by D(a) = {feA': \\f\\ = 1 =/(a)}. For a £ A the numerical range of a is defined by V(a) = {f(a):feD(l)} and the numerical radius of a by v(a) = sup{|A|: Ae V{a)}.We shall write Sp(a) and p(a) for the spectrum and spectral radius of a, respectively.Let X be a real or complex Banach space. B(X) is a unital Banach algebra with respect to | • | and so each operator T e B(X) has a numerical

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Finite-Dimensional Normed Linear Spaces with Numerical Index 1

Cited by 16 publications

References 2 publications

Real Banach Spaces with Numerical Index 1

Real Banach Spaces with Numerical Index 1

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

Mapping Theorems and the Numerical Radius

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