Linear evolution operators on spaces of periodic functions

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 $\mathcal{W}_{per}^{p}:=\{u\in W^{1,p}(0,2\pi ;X)\cap L^{p}(0,2\pi ;W):u(0)=u(2\pi)\}$ into $\mathcal{X} ^{p}:=L^{p}(0,2\pi ;X)$ when $p\in (1,\infty).$ The primary focus is on the Fredholmness and index of $D_{A},$ but a number of related issues are also discussed, 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 ensuring that $D_{A}$ is an isomorphism. Motivated by applications when $D_{A}$ arises as the linearization of a nonlinear operator, we also address similar questions in higher order spaces, which amounts to proving (nontrivial) regularity properties. Since we do not assume that $\pm A(t)$ generates any semigroup, approaches based on evolution systems are ruled out. In particular, we do not make use of any analog or generalization of Floquet's theory. Instead, some arguments, which rely on the autonomous case (for which results have only recently been made available) and a partition of unity, are more reminiscent of the methods used in elliptic PDE theory with variable coefficients.