Proceedings Mathematical Sciences | |
Isometric Multipliers of $L^p(G, X)$ | |
U B Tewari2  P K Chaurasia1  | |
[1] $$;Department of Mathematics, Indian Institute of Technology, Kanpur 0 0, India$$ | |
关键词: Locally compact group; Haar measure; Banach space-valued measurable functions; isometric multipliers.; | |
DOI : | |
学科分类:数学(综合) | |
来源: Indian Academy of Sciences | |
【 摘 要 】
Let ðº be a locally compact group with a fixed right Haar measure and ð‘‹ a separable Banach space. Let $L^p(G, X)$ be the space of ð‘‹-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 ≤ ð‘ < ∞, ð‘ ≠2, under the assumption that ð‘‹ is not the $l^p$-direct sum of two non-zero subspaces. In fact we prove that if 𑇠is an isometric left multiplier of $L^p(G, X)$ onto itself then there exists $a y in G$ and an isometry 𑈠of ð‘‹ 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 ðº is non-compact and ð‘‹ is not the $l^p$-direct sum of two non-zero subspaces. If ðº is a locally compact abelian group and ð» is a separable Hilbert space, we define $A^p(G, H)={fin l^1(G, H):hat{f}in L^p(ð›¤, H)}$ where 𛤠is the dual group of ðº. We characterize the isometric, invertible, left multipliers of $A^p(G, H)$, provided ðº is non-compact. Finally, we use the characterization of isometries of ð¶(ðº,ð‘‹) for ðº compact to determine the isometric left multipliers of ð¶(ðº,ð‘‹) provided ð‘‹* is strictly convex.
【 授权许可】
Unknown
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