Linear evolution operators on spaces of periodic functions
Wolfgang Arendt, Patrick J. Rabier · Communications on Pure & Applied Analysis · 2008
Given a family $A(t)$ of closed unbounded operators on a UMD Banach space $X$with common domain $W,$ we investigate various properties of the operator$D_{A}:=\frac{d}{dt}-A(\cdot)$ acting from $\mathcal{W}_{per}^{p}:=\{u\inW^{1,p}(0,2\pi ;X)\cap L^{p}(0,2\pi ;W):u(0)=u(2\pi)\}$ into $\mathcal{X}^{p}:=L^{p}(0,2\pi ;X)$ when $p\in (1,\infty).$ The primary focus is on theFredholmness and index of $D_{A},$ but a number of related issues are alsodiscussed, such as the independence of the index and spectrum of $D_{A}$upon $p$ or upon the pair $(X,W)$ as well as sufficient conditions ensuringthat $D_{A}$ is an isomorphism. Motivated by applications when $D_{A}$arises as the linearization of a nonlinear operator, we also address similarquestions in higher order spaces, which amounts to proving (nontrivial)regularity properties. Since we do not assume that $\pm A(t)$ generates anysemigroup, approaches based on evolution systems are ruled out. Inparticular, we do not make use of any analog or generalization of Floquet'stheory. Instead, some arguments, which rely on the autonomous case (forwhich results have only recently been made available) and a partition ofunity, are more reminiscent of the methods used in elliptic PDE theory withvariable coefficients.