Ideal properties and integral extension of convolution operators on $L^\infty (G)$

Abstract

We investigate operator ideal properties of convolution operators $C_\lambda $ (via measures $\lambda$) acting in ${L^\infty (G)}$, with $G$ a compact abelian group. Of interest is when $C_\lambda$ is compact, as this corresponds to $\lambda$ having an integrable density relative to Haar measure $\mu$, i.e., $\lambda \ll \mu $. Precisely then is there an \textit{optimal} Banach function space $L^1 (m_\lambda)$ available which contains ${L^\infty (G)}$ properly, densely and continuously and such that $C_\lambda$ has a continuous, ${L^\infty (G)}$-valued, linear extension $I_{m_\lambda}$ to $L^1 (m_\lambda)$. A detailed study is made of $L^1 (m_\lambda)$ and $I_{m_\lambda}$. Amongst other things, it is shown that $C_\lambda$ is compact iff the finitely additive, ${L^\infty (G)}$-valued set function $m_\lambda (A) := C_\lambda ({\chi_{_{_{\scriptstyle{A}}}}})$ is norm $\sigma$-additive iff $\lambda \in L^1 (G)$, whereas the corresponding optimal extension $I_{m_\lambda}$ is compact iff $\lambda \in C (G)$ iff $m_\lambda$ has finite variation. We also characterize when $m_\lambda$ admits a Bochner (resp.\ Pettis) $\mu$-integrable, $L^{\infty} (G)$-valued density.