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arXiv math/0503098math.FA
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$L^p(G, X)$的等距乘子

Isometric multipliers of $L^p(G, X)$

U B Tewari, P K Chaurasia

备注：
7 pages
发表信息：
Proc. Indian Acad. Sci. (Math. Sci.), Vol. 115, No. 1, February 2005, pp. 103-109
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摘要：

设$G$是具有固定右Haar测度的局部紧群，$X$是可分Banach空间.设$L^p(G,X)$是$X$值可测函数空间，其范数函数在通常的$L^{p}$中. $L^p(G,X)$的左乘子是$ L^p(G,X)$上的有界线性算子，它与所有左平移可交换。我们使用$L^p(G,X)$到自身上的等距的表征来表征$L^p(G,X)$对于$1\leq p <\infty$，$p\neq 2$的等距的，可逆的，左乘子，在假设$X$不是两个非零子空间的$\ell^p$直和的情况下。事实上，我们证明，如果$T$是一个等距左乘$L^p(G,X)$到自己，那么存在一个$y \in G$和等距$X$到自己的$U$，使$Tf(x)= U(R_yf)(x)$。作为应用，我们确定了$L^1 \bigcap L^p (G,X)$和$L^1 \bigcap C_0 (G,X)$的等距左乘子，其中$G$是非紧的，$X$不是两个非零子空间的$\ell^p$直和.若$G$是局部紧交换群，$H$是可分Hilbert空间，我们定义了$A^p (G,H) = \{f\in L^1(G,H)\hbox{:} \hat{f}\in L^p(Γ,H)\}$，其中$Γ$是$G$的对偶群.我们刻画了$A^p (G,H)$的等距的、可逆的、左乘子，只要$G$是非紧的。最后，我们利用$C(G,X)$的等距性对$G$紧的刻画，在$X^*$是严格凸的条件下，确定了$C(G,X)$的等距左乘子。

英文摘要：

Let $G$ be a locally compact group with a fixed right Haar measure and $X$ a separable Banach space. Let $L^p(G,X)$ be the space of $X$-valued measurable functions whose norm-functions are in the usual $L^{p}$. A left multiplier of $L^p(G,X)$ is a bounded linear operator on $ L^p(G,X)$ which commutes with all left translations. We use the characterization of isometries of $L^p(G,X)$ onto itself to characterize the isometric, invertible, left multipliers of $L^p(G,X)$ for $1\leq p <\infty$, $p\neq 2$, under the assumption that $X$ is not the $\ell^p$-direct sum of two non-zero subspaces. In fact we prove that if $T$ is an isometric left multiplier of $L^p(G,X)$ onto itself then there exists a $y \in G$ and an isometry $U$ of $X$ onto itself such that $Tf(x)= U(R_yf)(x)$. As an application, we determine the isometric left multipliers of $L^1 \bigcap L^p (G,X)$ and $L^1 \bigcap C_0 (G,X)$ where $G$ is non-compact and $X$ is not the $\ell^p$-direct sum of two non-zero subspaces. If $G$ is a locally compact abelian group and $H$ is a separable Hilbert space, we define $A^p (G,H) = \{f\in L^1(G,H)\hbox{:} \hat{f}\in L^p(Γ,H)\}$ where $Γ$ is the dual group of $G$. We characterize the isometric, invertible, left multipliers of $A^p (G,H)$, provided $G$ is non-compact. Finally, we use the characterization of isometries of $C(G,X)$ for $G$ compact to determine the isometric left multipliers of $C(G,X)$ provided $X^*$ is strictly convex.

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