Numerical Ranges II

Bonsall, F. F.; Duncan, John F. R.

doi:10.1017/cbo9780511662515

Cited by 383 publications

(210 citation statements)

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“…In the complex case, it is a celebrated result due to H. Bohnenblust and S. Karlin [2] (see also [8]) that n(X) 1/ e, so the numerical radius is always an equivalent norm. Classical references on this topic are the monographs by F. Bonsall and J. Duncan [3,4]. For recent results we refer the reader to [6,10,11,15,16,18] and the survey paper [13].…”

Section: It Is Clear That V Is a Seminorm On L(x) Satisfying V(t )mentioning

confidence: 99%

Finite-dimensional Banach spaces with numerical index zero

Martı́n

Merí

Rodríguez-Palacios

2004

Indiana Univ. Math. J.

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Abstract. We prove that a finite-dimensional Banach space X has numerical index 0 if and only if it is the direct sum of a real space X 0 and nonzero complex spaces X 1 , . . . , Xn in such a way that the equalityholds for suitable positive integers q 1 , . . . , qn, and every ρ ∈ R and every x j ∈ X j (j = 0, 1, . . . , n). If the dimension of X is two, then the above result gives X = C, whereas dim(X) = 3 implies that X is an absolute sum of R and C. We also give an example showing that, in general, the number of complex spaces cannot be reduced to one.

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Section: It Is Clear That V Is a Seminorm On L(x) Satisfying V(t )mentioning

confidence: 99%

Finite-dimensional Banach spaces with numerical index zero

Martı́n

Merí

Rodríguez-Palacios

2004

Indiana Univ. Math. J.

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“…D NOTE 1. By Lemma 1 and the condition (3) in Theorem 1, the proof of Theorem 1 can be reduced to the case in which T satisfies the following additional condition: [3] An invariant subspace theorem 13 LEMMA 2 .…”

Section: For / € H°°(g) and A € G We Deonte By F\ The Unique Functionmentioning

confidence: 99%

An invariant subspace theorem on subdecomposable operators

Liu¹

2000

Bull. Austral. Math. Soc.

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raised a conjecture on the invariant subspace problem in 1994. In this paper, we prove the conjecture under an additional condition, and obtain an invariant subspace theorem on subdecomposable operators.In [9] Mohebi and Radjabalipour raised the following conjecture. (1) qT = Bq for some injective q € B(X, Z) with a closed range qX.(2) There exist sequences {G(n)} of open sets andIt is easy to see that the Mohebi-Radjabalipour Conjecture, if true, will contain the main results of [1,2,4,5,7,8,9] (and others) as special cases.In the present article, using the S. Brown Technique, we prove the MohebiRadjabalipour Conjecture under an additional condition. But the additional condition will be used in only one place, namely in the proof of Lemma 4. Our main result is as follows. (4) {qX + M(n)} is a sequence of closed sets in Z.

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“…In the complex case, it is a deep result that n(X) is always bigger or equal than 1/e. Classical references here are the monographs from the 1970's [2,3]. The state-of-the-art on numerical indices may be found in the recent survey [9] and references therein.…”

Section: Introductionmentioning

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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Numerical Ranges II

Cited by 383 publications

References 0 publications

Finite-dimensional Banach spaces with numerical index zero

Finite-dimensional Banach spaces with numerical index zero

An invariant subspace theorem on subdecomposable operators

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

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