Mapping properties of operator-valued pseudo-differential operators

In this paper, we investigate the mapping properties of pseudo-differential operators with operator-valued symbols. We prove the boundedness of regular symbols on Sobolev spaces H-2(alpha) (R-d; L-2 (M)) and Besov spaces B-p, q(alpha) (R-d; L-p(M)) for alpha is an element of R and 1 <= p, q <= infinity, as well as the boundedness of forbidden symbols on H-2(alpha) (R-d; L-2(M)) and B-p, q(alpha)(R-d; L-p(M)) for alpha > 0 and 1 <= p, q <= infinity. Thanks to the smooth atomic decomposition of the operator-valued Triebel-Lizorkin spaces F-1(alpha, c)(R-d, M) obtained in our previous paper, we establish the F-1(alpha, c)-regularity of regular symbols for every alpha is an element of R, and the F-1(alpha, c)-regularity of forbidden symbols for alpha > 0. As applications, we obtain the same results on the usual and quantum tori. (C) 2019 Published by Elsevier Inc.