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 $$B_{pq}^s({\mathbb T}^n,E)$$ and on the Sobolev space $$W_p^k({\mathbb T}^n,E)$$ , where E is an arbitrary Banach space, $$1\le p,q\le \infty $$ , $$s\in {\mathbb R}$$ and $$k\in {\mathbb N}_0$$ . 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.