Ideal properties and integral extension of convolution operators on \(L^\infty (G)\) (Q2902318)

The authors investigate certain properties of the convolution operator \(C_\lambda(f)=f*\lambda\) between measures \(\lambda\) and functions \(f\in L^\infty(G)\) where \(G\) is a compact abelian group in terms of properties of the measure \(\lambda\). It is known that \(C_\lambda\) has a linear extension \(I_{m_\lambda}\) to \(L^1(m_\lambda)\) where \(m_\lambda\) is an \(L^\infty(G)\)-valued finitely additive measure and \(L^1(m_\lambda)\) corresponds to the optimal domain of \(C_\lambda\). Properties such as the compactness of \(C\lambda\) correspond to \(\lambda<<\mu\), i.e. \(\lambda\in L^1(G)\) (where \(\mu\) is the Haar measure on the group) or equivalently to the fact that \(m_\lambda\) is countably additive. However the compactness of the extension \(I_{m_\lambda}\) corresponds to the case where \(m_\lambda\) is of finite variation (equivalently \(\lambda\in C(G)\)). When restricting to the case \(d\lambda=g d\mu\) some connections between properties of \(g\) and properties of the operator \(C_\lambda\) are analyzed. For instance \(L^p(G)\subset L^1(m_\lambda)\) is equivalent to \(g\in L^{p'}(G)\) or the operator \(C_\lambda\in \Pi_p(L^\infty(G), L^\infty(G))\), i.e. \(C_\lambda\) is a \(p\)-summing operator. Certain intrinsic properties of the measure \(m_\lambda\) lead to a certain structure in the optimal domain and the associated integration operator. The case where \(m_\lambda\) has a Gelfand, Pettis or Bochner integrable density with respect to \(\mu \) is analyzed.