arXiv math/94022051994-02-08math.FA

关于l_p空间的张量积的结构

On the structure of tensor products of l_p spaces

Alvaro Arias, Jeff Farmer

摘要：

本文研究了$\ell_p$ -空间（投影、补子空间、自反、同构等）的（内射和投射）张量积的一些结构性质。我们结合这些结果与组合参数，以解决这些空间和它们的素数的问题。我们的主要结果是：\medbreak \item{(1)}如果$1<p<\infty$，那么$B(\ell_p)\approx B(L_p)$（$B(X)$由$X$上的有界线性算子组成）。\medbreak \item{(2)}如果每个$i\neq j$都有${1\over p_i}+{1\over p_j}\leq1$，或者如果所有的$p_i$都相等，那么$\ell_{p_1}\hat{\otimes}\cdots \hat{\otimes}\ell_{p_N}$是主的. \medbreak \item{(3)} $\ell_p$嵌入到$\ell_{p_1}\hat{\otimes}\cdots \hat{\otimes}\ell_{p_N}$中当且仅当存在$A\subset \{1,2,\cdots,n\}$使得${1\over p}=\min\{\sum_{i\in A}{1\over p_i},1\}$。如果$1\leq p<\infty$和$m\geq1$，则齐次解析多项式空间${\cal P}_m(\ell_p)$和$m$的对称张量积是准素的。

英文摘要：

We examine some structural properties of (injective and projective) tensor products of $\ell_p$-spaces (projections, complemented subspaces, reflexivity, isomorphisms, etc.). We combine these results with combinatorial arguments to address the question of primarity for these spaces and their duals. Our main results are: \medbreak \item{(1)} If $1<p<\infty$, then $B(\ell_p)\approx B(L_p)$ ($B(X)$ consists of the bounded linear operators on $X$). \medbreak \item{(2)} If ${1\over p_i}+{1\over p_j}\leq1$ for every $i\neq j$, or if all of the $p_i$'s are equal, then $\ell_{p_1}\hat{\otimes}\cdots \hat{\otimes}\ell_{p_N}$ is primary. \medbreak \item{(3)} $\ell_p$ embeds into $\ell_{p_1}\hat{\otimes}\cdots \hat{\otimes}\ell_{p_N}$ if and only if there exists $A\subset \{1,2,\cdots,n\}$ such that ${1\over p}=\min\{\sum_{i\in A}{1\over p_i},1\}$. \medbreak \item{(4)} If $1\leq p<\infty$ and $m\geq1$, then the space of homogeneous analytic polynomials ${\cal P}_m(\ell_p)$ and the symmetric tensor product of $m$ copies of $\ell_p$ are primary.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文