Generation of Semigroups for Vector-Valued Pseudodifferential Operators on the Torus

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. Here, we restrict ourselves to pseudodifferential operators with x-independent symbols (Fourier multipliers). We show that a parabolic toroidal pseudodifferential operator generates an analytic semigroup on the Besov space B^s_pq(Tⁿ,E) and on the Sobolev space W^k_p(Tⁿ,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 parameter-dependent sum. An application to abstract non-autonomous periodic pseudodifferential Cauchy problems gives the existence and uniqueness of classical solutions for such problems.