  In this paper, we consider pseudodifferential operators on the torus with
operator-valued symbols and prove continuity properties on vector-valued
toroidal Besov spaces, without assumptions on the underlying Banach spaces. The
symbols are of limited smoothness with respect to $x$ and satisfy a finite
number of estimates on the discrete derivatives. The proof of the main result
is based on a description of the operator as a convolution operator with a
kernel representation which is related to the dyadic decomposition appearing in
the definition of the Besov space.
