# American Institute of Mathematical Sciences

January  2009, 8(1): 5-36. doi: 10.3934/cpaa.2009.8.5

## Linear evolution operators on spaces of periodic functions

 1 Abteilung Angewante Analysis, Universität Ulm, 89069 Ulm, Germany 2 Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, United States

Received  March 2008 Revised  August 2008 Published  October 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 $\mathcal{W}_{per}^{p}:=\{u\in W^{1,p}(0,2\pi ;X)\cap L^{p}(0,2\pi ;W):u(0)=u(2\pi)\}$ 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.
Citation: Wolfgang Arendt, Patrick J. Rabier. Linear evolution operators on spaces of periodic functions. Communications on Pure and Applied Analysis, 2009, 8 (1) : 5-36. doi: 10.3934/cpaa.2009.8.5
 [1] Nguyen Dinh Cong, Roberta Fabbri. On the spectrum of the one-dimensional Schrödinger operator. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 541-554. doi: 10.3934/dcdsb.2008.9.541 [2] Yong-Kum Cho. A quadratic Fourier representation of the Boltzmann collision operator with an application to the stability problem. Kinetic and Related Models, 2012, 5 (3) : 441-458. doi: 10.3934/krm.2012.5.441 [3] Christoph Bandt, Helena PeÑa. Polynomial approximation of self-similar measures and the spectrum of the transfer operator. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4611-4623. doi: 10.3934/dcds.2017198 [4] Yucheng Bu, Yujun Dong. Existence of solutions for nonlinear operator equations. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4429-4441. doi: 10.3934/dcds.2019180 [5] Lilun Zhang, Le Li, Chuangxia Huang. Positive stability analysis of pseudo almost periodic solutions for HDCNNs accompanying $D$ operator. Discrete and Continuous Dynamical Systems - S, 2022, 15 (7) : 1651-1667. doi: 10.3934/dcdss.2021160 [6] Melvin Faierman. Fredholm theory for an elliptic differential operator defined on $\mathbb{R}^n$ and acting on generalized Sobolev spaces. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1463-1483. doi: 10.3934/cpaa.2020074 [7] Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Inverse problems for evolution equations with time dependent operator-coefficients. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 737-744. doi: 10.3934/dcdss.2016025 [8] Zhaoqiang Ge. Controllability and observability of stochastic implicit systems and stochastic GE-evolution operator. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 339-351. doi: 10.3934/naco.2021009 [9] Earl Berkson. Fourier analysis methods in operator ergodic theory on super-reflexive Banach spaces. Electronic Research Announcements, 2010, 17: 90-103. doi: 10.3934/era.2010.17.90 [10] Lianwang Deng. Local integral manifolds for nonautonomous and ill-posed equations with sectorially dichotomous operator. Communications on Pure and Applied Analysis, 2020, 19 (1) : 145-174. doi: 10.3934/cpaa.2020009 [11] Zhengxin Zhou. On the Poincaré mapping and periodic solutions of nonautonomous differential systems. Communications on Pure and Applied Analysis, 2007, 6 (2) : 541-547. doi: 10.3934/cpaa.2007.6.541 [12] Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, Dušan D. Repovš. Periodic solutions for implicit evolution inclusions. Evolution Equations and Control Theory, 2019, 8 (3) : 621-631. doi: 10.3934/eect.2019029 [13] Feride Tığlay. Integrating evolution equations using Fredholm determinants. Electronic Research Archive, 2021, 29 (2) : 2141-2147. doi: 10.3934/era.2020109 [14] Filippo Gazzola. On the moments of solutions to linear parabolic equations involving the biharmonic operator. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3583-3597. doi: 10.3934/dcds.2013.33.3583 [15] Ai-Ling Yan, Gao-Yang Wang, Naihua Xiu. Robust solutions of split feasibility problem with uncertain linear operator. Journal of Industrial and Management Optimization, 2007, 3 (4) : 749-761. doi: 10.3934/jimo.2007.3.749 [16] Phuong Le. Liouville theorems for stable weak solutions of elliptic problems involving Grushin operator. Communications on Pure and Applied Analysis, 2020, 19 (1) : 511-525. doi: 10.3934/cpaa.2020025 [17] Pasquale Candito, Giovanni Molica Bisci. Multiple solutions for a Navier boundary value problem involving the $p$--biharmonic operator. Discrete and Continuous Dynamical Systems - S, 2012, 5 (4) : 741-751. doi: 10.3934/dcdss.2012.5.741 [18] Foued Mtiri. Liouville type theorems for stable solutions of elliptic system involving the Grushin operator. Communications on Pure and Applied Analysis, 2022, 21 (2) : 541-553. doi: 10.3934/cpaa.2021187 [19] Caiping Liu, Heungwing Lee. Lagrange multiplier rules for approximate solutions in vector optimization. Journal of Industrial and Management Optimization, 2012, 8 (3) : 749-764. doi: 10.3934/jimo.2012.8.749 [20] Leszek Gasiński, Nikolaos S. Papageorgiou. Periodic solutions for nonlinear nonmonotone evolution inclusions. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 219-238. doi: 10.3934/dcdsb.2018015

2021 Impact Factor: 1.273