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

[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 ≤ 𝑝 < ∞, 𝑝 ≠ 2, under the assumption that 𝑋 is not the $l^p$-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 $Tf(x) = U(R_y f)(x)$. As an application, we determine the isometric left multipliers of $L^1 cap L^p(G, X)$ and $L^1 cap C_0(G, X)$ where 𝐺 is non-compact and 𝑋 is not the $l^p$-direct sum of two non-zero subspaces. If 𝐺 is a locally compact abelian group and 𝐻 is a separable Hilbert space, we define $A^p(G, H)={fin l^1(G, H):hat{f}in L^p(𝛤, H)}$ 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.

[学科分类] 数学（综合）

[关键词] Locally compact group;Haar measure;Banach space-valued measurable functions;isometric multipliers.