Abstract. 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 on the Besov space B s pq (T n , E) and on the Sobolev space W k p (T n , E), where E is an arbitrary Banach space, 1 ≤ p, q ≤ ∞, s ∈ R and k ∈ N0. For the proof of the Sobolev space result, we establish a uniform estimate on the kernel which is given as an infinite parameterdependent sum. An application to abstract non-autonomous periodic pseudodifferential Cauchy problems gives the existence and uniqueness of classical solutions for such problems.

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We consider operator-valued boundary value problems in (0, 2π) n with periodic or, more generally, ν-periodic boundary conditions. Using the concept of discrete vector-valued Fourier multipliers, we give equivalent conditions for the unique solvability of the boundary value problem. As an application, we study vector-valued parabolic initial boundary value problems in cylindrical domains (0, 2π) n × V with ν-periodic boundary conditions in the cylindrical directions. We show that under suitable assumptions on the coefficients, we obtain maximal L q -regularity for such problems.2010 Mathematics Subject Classification. 35J40, 35K46.

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