Convolution Operators that Factor Through a Hilbert Space

Abstract

Let G be a compact abelian group, and let X be a Banach space. We investigate vector-valued extensions of a result of Pisier concerning certain convolution operators that factor through Hilbert spaces. For example we show that X is finite dimentional if and only if for any continuous X-valued function Φ defined on G, the operator convolution by Φ, λ → Φ*λ, from M(G), the space of regular Borel measures on G, into C(G,X), the space of continuous X-valued functions defined on G, factors through a Hilbert space if and only if the series of Fourier coefficients ∑ _γ∈Γ Φ(γ) is absolutely convergent in X.