Translation Invariant Operators and Multipliers of Banach-Valued Function Spaces

Abstract

Abstract Let G be a locally compact abelian group, and A be a commutative Banach algebra and X a Banach A-module. In this paper, we investigate the invariant operators from a Banach-valuad function space defined on G into another Banach-valued function space, and characterize the space of all invariant operators as the following isometrically isomorphic relations under some appropriate conditions: (i) (L 1 (G, Y), L 1 (G, X)) ≅ (Y, M(G, X)); (ii) (L 1 (G, Y), L P (G, X)) ≅ (Y, L P (G, X)), 1 ∞ (iii) Hom L 1 (G, A) (L 1 (G, A), L 1 ≅ Hom A (A, M (G, X)); (iv) Hom L 1 (G, A) (L 1 (G, A), L P (G, X)) ≅ Hom A (A, L p (G, X)), 1 ∞ where (E(G, Y), G(G, X)) denotes the space of all invariant operators of E to F, ≅(Y, Z) is the space of bounded linear operators from Y to Z, and Hom A means the A-module homomorphisms. Moreover (i) and (ii) with Y = A coincide with (iii) and (iv) respectively if and only if A the complex field. This means that any invariant operator of a Banach function space is a multiplier if and only if A ≅