Vol. 18, No. 3, 1966

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Quasimeasures and operators commuting with convolution

Garth Ian Gaudry

Vol. 18 (1966), No. 3, 461–476

Let G be a Hausdorff locally compact abelian group. In this paper we characterise completely those continuous linear operators T from Cc(G) (the space of continuous functions with compact supports endowed with the inductive limit topology) into M(G) (the space of measures with the vague topology of measures) which commute with convolution: T(f g) = (Tf) g. They are represented by convolution with a “quasimeasure”. As a corollary of this theorem, we have the result that the space of multipliers from Lp(G)(p) to Lq(G) is isomorphic to a subspace of the space of quasimeasures.

The quasimeasures are defined as the elements of the dual of a certain inductive limit of Banach spaces. We develop some of the theory of pseudomeasures and of quasimeasures and establish the structural relationship of quasimeasures to pseudomeasures.

Mathematical Subject Classification
Primary: 46.80
Secondary: 42.56
Received: 18 June 1965
Published: 1 September 1966
Garth Ian Gaudry