Integral extension of multiplier operators in A(G)

Abstract

We investigate operator ideal properties of multiplier operators Tϕ (via ϕ∈l∞(Γ) with Γ the dual group) acting in the space A(G) of all integrable functions on an infinite, compact abelian group G whose Fourier coefficients are absolutely summable. Of interest is when Tϕ is 1-summing, as this corresponds to \(\varphi= \widehat {f}\) with f square integrable relative to Haar measure. Precisely then there is an optimal Banach function space L1(mϕ) available which contains A(G) densely and continuously and such that Tϕ has a continuous A(G)-valued linear extension \(I_{m_{\varphi}}\) to L1(mϕ). Relevant for studying L1(mϕ) and \(I_{m_{\varphi}}\) is the space Sp, 1≤p≤∞, of all functions in Lp(G) whose Fourier series is unconditionally convergent in Lp(G). Amongst other things, it is shown that \(I_{m_{\varphi}}\) is 1-summing iff Tϕ is nuclear iff the vector measure \(m_{\varphi}(E) := T_{\varphi}( \chi _{{}_{E}}) \) has finite variation. Moreover, L1(mϕ) is a homogeneous Banach space. Hence, it is also a Banach algebra and an L1(G)-module under convolution.