Polynomial Grothendieck properties
Polynomial Grothendieck properties
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DOI：
10.1017/s0017089500031116
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发表时间：
1994-04
影响因子：
0.5
通讯作者：
Manuel González;J. Gutiérrez
中科院分区：
文献类型：
--
作者：
Manuel González;J. Gutiérrez
一个Banach空间sE具有Grothendieck性质，如果从E到c ~ 0的每个（线性有界）算子都是弱紧的.证明了对整数k > 1，从E到c 0的k-齐次多项式是弱紧的当且仅当E上的量值多项式空间（kE）是自反的.这等价于E的对称A>折叠投影张量积（即，（kE）的前对偶）具有Grothendieck性质。还刻画了射影张量积EF的Grothendieck性质。此外，E的Grothendieck性质被描述为多项式序列。最后，证明了如果从E到c 0的每个算子都是完全连续的，那么这些空间之间的每个多项式也是完全连续的。
Abstract A Banach space sE has the Grothendieck property if every (linear bounded) operator from E into c0 is weakly compact. It is proved that, for an integer k > 1, every k-homogeneous polynomial from E into c0 is weakly compact if and only if the space (kE) of scalar valued polynomials on E is reflexive. This is equivalent to the symmetric A>fold projective tensor product of £(i.e., the predual of (kE)) having the Grothendieck property. The Grothendieck property of the projective tensor product EF is also characterized. Moreover, the Grothendieck property of E is described in terms of sequences of polynomials. Finally, it is shown that if every operator from E into c0 is completely continuous, then so is every polynomial between these spaces.