Isometric multipliers of Lᵖ(G, X)

Original · EN

Let G be a locally compact group with a fixed right Haar measure and X a separable Banach space. Let Lᵖ(G,X) be the space of X-valued measurable functions whose norm-functions are in the usual Lᵖ. A left multiplier of Lᵖ(G,X) is a bounded linear operator on Lᵖ(G,X) which commutes with all left translations. We use the characterization of isometries of Lᵖ(G,X) onto itself to characterize the isometric, invertible, left multipliers of Lᵖ(G,X) for 1≤ p <∞, p≠ 2, under the assumption that X is not the ℓᵖ-direct sum of two non-zero subspaces. In fact we prove that if T is an isometric left multiplier of Lᵖ(G,X) onto itself then there exists a y ∈ G and an isometry U of X onto itself such that Tf(x)= U(Ryf)(x). As an application, we determine the isometric left multipliers of L¹ Lᵖ (G,X) and L¹ C₀ (G,X) where G is non-compact and X is not the ℓᵖ-direct sum of two non-zero subspaces. If G is a locally compact abelian group and H is a separable Hilbert space, we define Aᵖ (G,H) = {f∈ L¹(G,H): f∈ Lᵖ(Γ,H)} where Γ is the dual group of G. We characterize the isometric, invertible, left multipliers of Aᵖ (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.