多项式Grothendieck性质
Polynomial Grothendieck properties
摘要：
巴拿赫空间$E$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$c_0$的每个$k$ -齐次多项式是弱紧的当且仅当$E$上的标量值多项式空间${\cal P}(^kE)$是自反的.这等价于$E$的对称$k$ -折叠投影张量积（即，${\cal P}(^kE)$的前对偶）具有Grothendieck性质。还刻画了射影张量积$E\widehat{\bigotimes}F$的Grothendieck性质.此外，$E$的Grothendieck性质描述的多项式序列。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.
英文摘要：
A Banach space $E$ has the Grothendieck property if every (linear bounded) operator from $E$ into $c_0$ is weakly compact. It is proved that, for an integer $k>1$, every $k$-homogeneous polynomial from $E$ into $c_0$ is weakly compact if and only if the space ${\cal P}(^kE)$ of scalar valued polynomials on $E$ is reflexive. This is equivalent to the symmetric $k$-fold projective tensor product of $E$ (i.e., the predual of ${\cal P}(^kE)$) having the Grothendieck property. The Grothendieck property of the projective tensor product $E\widehat{\bigotimes}F$ 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 $c_0$ is completely continuous, then so is every polynomial between these spaces.
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