Abstract. Let G be a locally compact abelian group and A" be a Banach space. Let Ll(G, X) be the Banach space of X-valued Bochner integrable functions on G. We prove that the space of bounded linear translation invariant operators of L'(G, X) can be identified with L(X, M(G, X)), the space of bounded linear operators from X into M(G, X) where M(G, X) is the space of Jf-valued regular, Borel measures of bounded variation on G. Furthermore, if A is a commutative semisimple Banach algebra with identity of norm 1 then L\G, A) is a Banach algebra and we prove that the space of multipliers of L\G, A) is isometrically isomorphic to M{G, A). It also follows that if dimension of A is greater than one then there exist translationinvariant operators of Ll(G, A) which are not multipliers of L1(G, A).

Abstract. Let T be a multiplier of a Segal algebra S on a locally compact abelian group G. We prove that T2(S) is closed if and only if T is a product of an idempotent and an invertible multiplier. We also show that the techniques developed in the proof of this theorem can be used to obtain some other known results.

Abstract. Let T be a multiplier of a Segal algebra S on a locally compact abelian group G. We prove that T2(S) is closed if and only if T is a product of an idempotent and an invertible multiplier. We also show that the techniques developed in the proof of this theorem can be used to obtain some other known results.

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