Absolutely summing multiplier operators in $L^p (G)$ for $p > 2$

Abstract:

Let $G$ be an infinite compact abelian group. If its dual group $\Gamma$ contains an element of infinite order, then it is known that, for every $4<p<\infty$, there exists a function $g \in L^p (G)$ whose associated convolution operator $C_g : f \mapsto f * g$ (on $L^p (G)$) is absolutely summing but the Fourier series of $g$ fails to be unconditionally convergent to $g$ in $L^p (G)$. It is shown that the restriction on $\Gamma$ containing an element of infinite order can be removed and also that the range of $p$ can be extended to arbitrary $p \in (2, \infty )$.