Characterizations of nuclear pseudo-differential operators on S1 with applications to adjoints and products

We give necessary and sufficient conditions on the symbols to guarantee that the corresponding pseudo-differential operators are nuclear from L-p1 (S-1) into L-p2 ((S)1) for 1 <= p(1), p(2) <= infinity. Applications are given to adjoints of nuclear pseudo-differential operators from L-p'2 (S-1) into L-p'1 (S1) for 1 <= p(1), p(2) < infinity and products of nuclear pseudo-differential operators on L-p(S-1), 1 <= p < infinity.