• Isometric multipliers ofLp(G, X)

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/115/01/0103-0109

• # Keywords

Locally compact group; Haar measure; Banach space-valued measurable functions; isometric multipliers

• # Abstract

Let G be a locally compact group with a fixed right Haar measure andX a separable Banach space. LetLp(G, X) be the space of X-valued measurable functions whose norm-functions are in the usualLp. A left multiplier ofLp(G, X) is a bounded linear operator onBp(G, X) which commutes with all left translations. We use the characterization of isometries ofLp(G, X) onto itself to characterize the isometric, invertible, left multipliers ofLp(G, X) for 1 ≤p ∞,p ≠ 2, under the assumption thatX is not thelp-direct sum of two non-zero subspaces. In fact we prove that ifT is an isometric left multiplier ofLp(G, X) onto itself then there existsa y ∃ G and an isometryU ofX onto itself such thatTf(x) = U(Ryf)(x). As an application, we determine the isometric left multipliers of L1Lp(G, X) and L1C0(G, X) whereG is non-compact andX is not the lp-direct sum of two non-zero subspaces. If G is a locally compact abelian group andH is a separable Hubert space, we define $$A^p (G,H) = \{ f \in L^1 (G,H):\hat f \in L^p (\Gamma ,H)\}$$ where г is the dual group of G. We characterize the isometric, invertible, left multipliers ofAp(G, H), provided G is non-compact. Finally, we use the characterization of isometries ofC(G, X) for G compact to determine the isometric left multipliers ofC(G, X) providedX* is strictly convex.

• # Author Affiliations

1. Department of Mathematics, Indian Institute of Technology, Kanpur - 208 016, India

• # Proceedings – Mathematical Sciences

Volume 131, 2021
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