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

Abstract

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 $A^p (G,H) = \{f\in L^1(G,H)\hbox{:} \hat{f}\in L^p(\Gamma,H)\}$ where $\Gamma$ 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.