On Tsirelson’s space

Casazza, Peter G.; Johnson, William B.; Tzafriri, Lior

doi:10.1007/bf02760508

Cited by 113 publications

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“…Since X* is a space of type 2 having an unconditional basis, and does not contain a copy of l 2 (see [3]), the following result implies that A"* e W 2 . PROPOSITION Proof.…”

Section: Proof Consider the Operator T:b(l)-+x Defined By T(x E ) =mentioning

confidence: 97%

Remarks on the range of a vector measure

Castilo

Sánchez

1994

Glasgow Math. J.

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A long-standing problem is the characterization of subsets of the range of a vector measure. It is known that the range of a countably additive vector measure is relatively weakly compact and, in addition, possesses several interesting properties (see [2]). In [6] it is proved that if m : 2-•A' is a countably additive vector measure, then the range of m has not only the Banach-Saks property, but even the alternate Banach-Saks property. A tantalizing conjecture, which we shall disprove in this article, is that the range of m has to have, for some p > 1, the p-Banach-Saks property. Another conjecture, which has been around for some time (see [2]) and is also disproved in this paper, is that weakly null sequences in the range of a vector measure admit weakly-2-summable sub-sequences. In fact, we shall show a weakly null sequence in the range of a countably additive vector measure having, for every p < °°, no weakly-p-summable sub-sequences.

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“…Since X* is a space of type 2 having an unconditional basis, and does not contain a copy of l 2 (see [3]), the following result implies that A"* e W 2 . PROPOSITION Proof.…”

Section: Proof Consider the Operator T:b(l)-+x Defined By T(x E ) =mentioning

confidence: 97%

Remarks on the range of a vector measure

Castilo

Sánchez

1994

Glasgow Math. J.

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“…Let T be any version of the 2-convexified Tsirelson space, presented in [CS;P,Chapter 13]. To fix attention, we set, in the notation from [P], T -Xs , with 5 = 2_1/2.…”

Section: Weak Hilbert Spaces Without An Unconditional Basismentioning

confidence: 99%

On constructing Banach spaces with no unconditional basis

Komorowski¹

1994

Proc. Amer. Math. Soc.

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Abstract.There exists a weak Hilbert space with no unconditional basis.The aim of this note is to show a rather general construction of Banach spaces with no unconditional basis. As a corollary, one obtains a weak Hilbert space with no unconditional basis, thus answering a question raised some years ago by several authors (cf., e.g., [CS, P]). Our approach is based on techniques first introduced by Johnson, Lindenstrauss, and Schechtman in [JLS] for the study of the Kalton-Peck space [KP]. These techniques were refined b> Ketonen [K] and generalized further by Borzyszkowski [B]. The novelty of the present general approach consists of the use of a simple interpolation trick, which is nevertheless strong enough to allow for the construction of a weak Hilbert space which contains a subspace with no unconditional basis. The main constructionThe standard notation from the Banach space theory used throughout this paper can be found, for example, in [P, TJ].Let us recall that, if (Zfc)£L, is a family of finite-dimensional subspaces of a Banach space X, then the unconditional constant of (Zk)kx=i , denoted by unc(Zk)kxLl , is the infimum of numbers c > 0 such that, for all finite sequences of vectors (xk)k , with xk £ Zk , and for all choices of signs (ek)k , the following holds:If unc(Zk)kx'=l is finite, we call (Zk)k*Ll an unconditional decomposition. A basis (e,)^, in a Banach space X is 1-conditional if unc(span{e,}g[) = 1. If A is a set of positive integers, we denote span({e,}/6/1) in X by X\A .Throughout this paper we fix an interpolation functor which to any 0 < 6 < 1 and any interpolation couple of Banach spaces (Xo, X\), with norms || • ||o and || • ||! respectively, corresponds the space (X0, X\)e , with the norm || • \\e such that:

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“…By [6], if p -mf{q: T has lower q estimates}, then T contains (uniformly) disjointly supported vectors which are (uniformly) equivalent to the unit vector basis of lp. By the definition of T and Theorem 3.3 of [2] we must have/? = 1.…”

Section: In [8]mentioning

confidence: 99%

“…e /" -Jx e E ^'Rx = {'Rx)j e /, -f a^Rx)"!, where 'R is the transpose of R. Therefore, £" is a reflexive Banach space which does not contain any lp. For details on the theory of holomorphic functions in infinite dimensions we refer to S. Dineen [5] and for a general survey on properties of Tsirelson's space we refer to P. G. Casazza [2].…”

mentioning

confidence: 99%