Isometric multipliers ofL p (G, X)

Abstract

Let G be a locally compact group with a fixed right Haar measure andX a separable Banach space. LetL p (G, X) be the space of X-valued measurable functions whose norm-functions are in the usualL p . A left multiplier ofL p (G, X) is a bounded linear operator onB p (G, X) which commutes with all left translations. We use the characterization of isometries ofL p (G, X) onto itself to characterize the isometric, invertible, left multipliers ofL p (G, X) for 1 ≤p ∞,p ≠ 2, under the assumption thatX is not thel p -direct sum of two non-zero subspaces. In fact we prove that ifT is an isometric left multiplier ofL p (G, X) onto itself then there existsa y ∃ G and an isometryU ofX onto itself such thatTf(x) = U (R y f)(x). As an application, we determine the isometric left multipliers of L1 ∩L p (G, X) and L1 ∩C 0 (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 ofA p (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.