arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 0802.0476math.FAmath.OA
收藏

Hilbert空间、Banach空间和算子空间之间的复插值

Complex Interpolation between Hilbert, Banach and Operator spaces

Gilles Pisier

备注：
A detailed proof of Hernandez Theorem 4.6 has been added, as well as various minor improvements and clarifications
发表信息：
Memoirs Amer. Math. Soc. 208 (2010), no. 978
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

受Vincent Lafforgue的一个问题的启发，我们研究了满足以下性质的Banach空间$X$：存在一个趋于零的函数$\vp\to Δ_X(\vp)$和$\vp>0$，使得每一个具有$\|T\|\le \vp$的算子$T\colon L_2\to L_2$在$L_1$和$L_\infty$上同时压缩（即具有范数PDMATHAMR），在$L_2(X)$上必须具有范数$\le Δ_X(\vp)$.我们表明，$Δ_X(\vp)\in O(\vp^α)$的一些$α>0$当$X$是同构的一个子空间的超积的$θ$ -希尔伯特空间的一些$ θ>0$（见推论\ref{comcor4.3}），其中$θ$ -希尔伯特是指在一个稍微更一般的意义上比我们以前的文件\cite{P1}。令$B_{r}(L_2(μ))$是$L_2(μ)$上所有正则算子的空间。我们能够描述复插值空间\[ (B_{r}(L_2(μ), B(L_2(μ))^θ. \].我们证明了$T\colon L_2(μ)\to L_2(μ)$属于这个空间当且仅当对于任何$θ$-Hilbertian空间$X$，$T\otimes id_X$在$L_2(X)$上有界.更一般地，我们能够描述任何对$1\le p_0,p_1\le \infty$和$0<θ<1$的空间$$ (B(\ell_{p_0}), B(\ell_{p_1}))^θ{\rm or} (B(L_{p_0}), B(L_{p_1}))^θ$$。同样，给定一个局部紧阿贝尔群$G$，令$M(G)$（resp. $PM(G)$）是复杂测度的空间（分别为伪测度）在配备有通常范数$\|μ\|_{M(G)} = |μ|(G)$的$G$上（分别\[ \|μ\|_{PM(G)} = \sup\{|\hatμ(γ)| \big| γ\in\hat G\}). \]我们类似地描述插值空间$(M(G), PM(G))^θ$。这个结果的各种扩展和变体将被给出，例如$B(\ell_2)$上的Schur乘子和算子空间。

英文摘要：

Motivated by a question of Vincent Lafforgue, we study the Banach spaces $X$ satisfying the following property: there is a function $\vp\to Δ_X(\vp)$ tending to zero with $\vp>0$ such that every operator $T\colon L_2\to L_2$ with $\|T\|\le \vp$ that is simultaneously contractive (i.e. of norm $\le 1$) on $L_1$ and on $L_\infty$ must be of norm $\le Δ_X(\vp)$ on $L_2(X)$. We show that $Δ_X(\vp)\in O(\vp^α)$ for some $α>0$ iff $X$ is isomorphic to a quotient of a subspace of an ultraproduct of $θ$-Hilbertian spaces for some $ θ>0$ (see Corollary \ref{comcor4.3}), where $θ$-Hilbertian is meant in a slightly more general sense than in our previous paper \cite{P1}. Let $B_{r}(L_2(μ))$ be the space of all regular operators on $L_2(μ)$. We are able to describe the complex interpolation space \[ (B_{r}(L_2(μ), B(L_2(μ))^θ. \] We show that $T\colon L_2(μ)\to L_2(μ)$ belongs to this space iff $T\otimes id_X$ is bounded on $L_2(X)$ for any $θ$-Hilbertian space $X$. More generally, we are able to describe the spaces $$ (B(\ell_{p_0}), B(\ell_{p_1}))^θ{\rm or} (B(L_{p_0}), B(L_{p_1}))^θ$$ for any pair $1\le p_0,p_1\le \infty$ and $0<θ<1$. In the same vein, given a locally compact Abelian group $G$, let $M(G)$ (resp. $PM(G)$) be the space of complex measures (resp. pseudo-measures) on $G$ equipped with the usual norm $\|μ\|_{M(G)} = |μ|(G)$ (resp. \[ \|μ\|_{PM(G)} = \sup\{|\hatμ(γ)| \big| γ\in\hat G\}). \] We describe similarly the interpolation space $(M(G), PM(G))^θ$. Various extensions and variants of this result will be given, e.g. to Schur multipliers on $B(\ell_2)$ and to operator spaces.

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。

↑