Alternating procedures in uniformly smooth Banach spaces
Alternating procedures in uniformly smooth Banach spaces
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均匀光滑巴纳赫空间中的交替过程
DOI：
10.1090/s0002-9939-1988-0937842-4
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发表时间：
1988
期刊：
影响因子：
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通讯作者：
I. Assani
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文献类型：
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作者：
I. Assani
设E是一致光滑Banach空间，C是R+上的真实的连续严格增函数集，u（O）= 0.在每个u处，我们可以关联一个唯一的对偶映射J.：E E* 使得（Ji，x，x）= IIJxI IIIxII且IIJxII = p（IIxI）。本文证明了：如果Tn是E上的线性压缩序列，则序列T T2*. T，n JA Tn. T2 T1 x强收敛于E* 范数，对所有x在E中.特别地，如果E* 也是均匀平滑的，则对于C中的任何u和v，序列JT 1 T2*.. TnJATn * T.Tx收敛于E范数。这推广了M. Akcoglu和L. Sucheston [1].导论.设E是一致光滑的Banach空间，即，limt 0（（zlx + thll 11 x）/t）在x，（x 00）和h中一致存在.已知E的对偶E* 是一致凸的。因此，对于E > 0，存在6 > 0，使得如果lx + yl > 2 6，Lxv < 1，并且yIIYI < 1，则lx yII < E（x，y E E*）。我们用C表示定义在[0，+ox]上的证明u（0）= 0的真实的严格递增连续函数的集合。对于任意的光滑空间E，存在唯一的对偶映射J，对任意的，uEC，J，：EE * 满足下列条件：（i）对任意的E中的x，（Jzx，x）= IJvHx 1 Ilx 1;（ii）对任意的E中的x，11 Jx 11 =（11 IX 11）.若E是一致光滑的，则JH，在E（对于范数拓扑）到E*（具有范数拓扑）的有界集上是一致连续的。(For关于这些概念的参考文献见[4]）。在[5] G. C. Rota为L1和L？？引入了以下程序：积极的收缩他认为产品TrT 2 * T. T T1。在文献[2]中，我们证明了当算子Ti是Hilbert空间上的线性压缩算子时，这类乘积的范数收敛性。当Banach空间E不是Hilbert空间时，T和T* 不作用于同一空间。为了使用乘积T T2，似乎有必要引入一个对偶映射JH。TN.我们的第一个结果是产品TT 2 * 的范数收敛。Tn J，Tn Tn...在一致光滑Banach空间E中，对任意对偶映射和任意线性压缩序列Ti，给出了Tx在E* 范数下的一个结果。利用在E* 上存在一个等价范数使得E* 是一致光滑的这一事实（见[4]），我们可以定义J，* 是E* 上的对偶映射，对C中的任意v。然后证明了积J，T ~ 2 T在E范数下的范数收敛性。T * T.对于E中的任意x，这推广了M. Akcoglu和L. Sucheston [1]在LP和特殊对偶映射中得到了证明。编辑于1988年2月19日收到。1980年数学学科分类（1985年修订）。小学28 D99，47 A35。
Let E be a uniformly smooth Banach space and C the set of real continuous strictly increasing functions ,u on R+ such that ,u(O) = 0. At each ,u we can associate a unique duality map J.:E E* such that (J;,x,x) = IIJxI III xII and IIJx II = p (IIx I). We prove in this note that if Tn is a sequence of linear contractions on E the sequence T T2* ... T,n JA Tn ... T2 T1 x converges strongly in E* norm for all x in E. In particular if E* is also uniformly smooth then for any ,u and v in C the sequence JT1 T2* ... TnJATn * T.Tx converges in E norm. This generalizes a result of M. Akcoglu and L. Sucheston [1]. Introduction. Let E be a uniformly smooth Banach space, i.e., limt0((zlx + thll 11x)/t) exists uniformly in x, (x 0 0) and h. It is known that the dual E* of E is then uniformly convex. So for E > 0 there exists 6 > 0 such that if lx + yl > 2 6, Lxv < 1, and yIIYI < 1, then lx yII < E (x,y E E*). We denote by C the set of real strictly increasing continuous functions verifying u(0) = 0 defined on [0, +ox[. The space E being smooth, there exists a unique duality map J, for any ,u E C, J,: E E* satisfying the following conditions (i) for any x in E, (Jzx,x) = I JvHx1 Ilxl ; (ii) for any x in E, 11Jx11 = (11IX11). If E is uniformly smooth then JH, is uniformly continuous on the bounded sets of E (for the norm topology) to E* (with the norm topology). (For a reference on these notions see [4]). In [5] G. C. Rota introduced the following procedure for L1 and L?? positive contractions. He considered the products TrT2* T... T T1. We showed in [2] the norm convergence of such products when the operators Ti are linear contractions on a Hilbert space. When the Banach space E is not a Hilbert space, T and T* do not act on the same space. It seems then necessary to introduce a duality map JH in order to use the product T T2. Tn. Our first result is the norm convergence of the products TT2* ... Tn J,TnTn... Tx in E* norm for any duality map and any sequence Ti of linear contractions in the uniformly smooth Banach space E. Using the fact (see [4]) that there exists an equivalent norm on E* for which E* is uniformly smooth, we can define J,* a duality map on E* for any v in C. We prove then the norm convergence in E norm of the products J,T T2T .. Tn J,,T * T. x for any x in E. This extends the result of M. Akcoglu and L. Sucheston [1] proved in LP and for particular duality maps. Received by the editors February 19, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 28D99, 47A35.