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 H₂ ^α(R^d;L₂(M)) and Besov spaces B_p,q ^α(R^d;L_p(M)) for α∈R and 1≤p,q≤∞, as well as the boundedness of forbidden symbols on H₂ ^α(R^d;L₂(M)) and B_p,q ^α(R^d;L_p(M)) for α>0 and 1≤p,q≤∞. Thanks to the smooth atomic decomposition of the operator-valued Triebel-Lizorkin spaces F₁ ^α,c(R^d,M) obtained in our previous paper, we establish the F₁ ^α,c-regularity of regular symbols for every α∈R, and the F₁ ^α,c-regularity of forbidden symbols for α>0. As applications, we obtain the same results on the usual and quantum tori.