Characterizations of nuclear pseudo-differential operators on $${\mathbb {S}}^1$$ S 1 with applications to adjoints and products

Abstract

We give necessary and sufficient conditions on the symbols to guarantee that the corresponding pseudo-differential operators are nuclear from \(L^{p_1}({\mathbb {S}}^1)\) into \(L^{p_2}({\mathbb {S}}^1)\) for \(1\le p_1,p_2<\infty \). Applications are given to adjoints of nuclear pseudo-differential operators from \(L^{p_2'}({\mathbb {S}}^1)\) into \(L^{p_1'}({\mathbb {S}}^1)\) for \(1\le p_1,p_2<\infty \) and products of nuclear pseudo-differential operators on \(L^p({\mathbb {S}}^1),\,1\le p<\infty \).