Communications on Pure and Applied Analysis (CPAA)

Linear evolution operators on spaces of periodic functions

Pages: 5 - 36, Volume 8, Issue 1, January 2009      doi:10.3934/cpaa.2009.8.5

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Wolfgang Arendt - Abteilung Angewante Analysis, Universit├Ąt Ulm, 89069 Ulm, Germany (email)
Patrick J. Rabier - Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, United States (email)

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.

Keywords:  Fredholm operator, spectrum, Fourier multiplier, nonautonomous evolution operator, periodic solutions.
Mathematics Subject Classification:  47A53, 45M15, 42A45.

Received: March 2008;      Revised: August 2008;      Available Online: October 2008.