  We consider toroidal pseudodifferential operators with operator-valued
symbols, their mapping properties and the generation of analytic semigroups on
vector-valued Besov and Sobolev spaces. We show that a parabolic toroiodal
pseudodifferential operator generates an analytic semigroup both on Banach
space-valued Besov spaces and on Banach space-valued Sobolev spaces. Here, no
condition on the Banach space is imposed. For the proof of the Sobolev space
result, we establish a uniform estimate on the kernel which is given as an
infinite parameter-dependent sum. An application to abstract non-autonomous
periodic pseudodifferential Cauchy problems gives the existence and uniqueness
of classical solutions for such problems.
