论文标题
在小的Grothendieck Theorem中的最佳性
On optimality of constants in the Little Grothendieck Theorem
论文作者
论文摘要
我们探讨了常数的最佳性，使得最近确定的几乎没有Grothendieck的不平等现象JB $^*$ - Triples和JB $^*$ - 代数。在我们的主要结果中，我们证明，对于每个有限的线性运算符$ t $，来自jb $^*$ - 代数$ b $进入复杂的Hilbert Space $ h $和$ \ VAREPSILON> 0 $，在B^*$中有一个规范的功能$φ\ $ \ | tx \ | \ le（\ sqrt {2}+\ varepsilon）\ | | t \ | \ | \ | \ | \ | x \ |_φ\ quad \ quad \ mbox {for} for} x \ inb。$ $ $ $ $ $ $ $ $ $ $ $我们还提供一个简单的例子，目睹该常数不能严格小于$ \ sqrt2 $，因此我们的主要定理是“渐近最佳”。对于I型JBW $^*$ - 代数，我们建立了正常功能的规范分解，可用于证明此特殊情况下的主要结果，并且似乎也具有独立的兴趣。作为一种工具，我们证明了希尔伯特空间上紧凑型操作员的施密特表示的可测量版本。
We explore the optimality of the constants making valid the recently established Little Grothendieck inequality for JB$^*$-triples and JB$^*$-algebras. In our main result we prove that for each bounded linear operator $T$ from a JB$^*$-algebra $B$ into a complex Hilbert space $H$ and $\varepsilon>0$, there is a norm-one functional $φ\in B^*$ such that $$\|Tx\|\le(\sqrt{2}+\varepsilon)\|T\|\|x\|_φ\quad\mbox{ for }x\in B.$$ The constant appearing in this theorem improves the best value known up to date (even for C$^*$-algebras). We also present an easy example witnessing that the constant cannot be strictly smaller than $\sqrt2$, hence our main theorem is `asymptotically optimal'. For type I JBW$^*$-algebras we establish a canonical decomposition of normal functionals which may be used to prove the main result in this special case and also seems to be of an independent interest. As a tool we prove a measurable version of the Schmidt representation of compact operators on a Hilbert space.
