Let G be a compact metrizable abelian group, and let X be a Banach space. We characterize convolution operators associated with a regular Borel X-valued measure of bounded semivariation that are compact (resp; weakly compact) from L~1(G), the space of integrable functions on G into L~1(G) X, the injective tensor product of L~1(G) and X. Along the way we prove a Fourier Convergence theorem for vector measures of relatively compact range that are absolutely continuous with respect to the Haar measure.
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