
Abstract
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.
Mathematics Subject Classification: 47A53, 45M15, 42A45.
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