American Institute of Mathematical Sciences

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

Linear evolution operators on spaces of periodic functions

Wolfgang Arendt 1, and  Patrick J. Rabier 2,

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.
Keywords: spectrum, Fourier multiplier, Fredholm operator, nonautonomous evolution operator, periodic solutions..
Mathematics Subject Classification: 47A53, 45M15, 42A4.
Citation: Wolfgang Arendt, Patrick J. Rabier. Linear evolution operators on spaces of periodic functions. Communications on Pure & 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 & 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 & 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 & 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 & Continuous Dynamical Systems, 2019, 39 (8) : 4429-4441. doi: 10.3934/dcds.2019180
[5]

Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Inverse problems for evolution equations with time dependent operator-coefficients. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 737-744. doi: 10.3934/dcdss.2016025
[6]

Zhaoqiang Ge. Controllability and observability of stochastic implicit systems and stochastic GE-evolution operator. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021009
[7]

Melvin Faierman. Fredholm theory for an elliptic differential operator defined on $ \mathbb{R}^n $ and acting on generalized Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1463-1483. doi: 10.3934/cpaa.2020074
[8]

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
[9]

Lianwang Deng. Local integral manifolds for nonautonomous and ill-posed equations with sectorially dichotomous operator. Communications on Pure & Applied Analysis, 2020, 19 (1) : 145-174. doi: 10.3934/cpaa.2020009
[10]

Zhengxin Zhou. On the Poincaré mapping and periodic solutions of nonautonomous differential systems. Communications on Pure & Applied Analysis, 2007, 6 (2) : 541-547. doi: 10.3934/cpaa.2007.6.541
[11]

Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, Dušan D. Repovš. Periodic solutions for implicit evolution inclusions. Evolution Equations & Control Theory, 2019, 8 (3) : 621-631. doi: 10.3934/eect.2019029
[12]

Feride Tığlay. Integrating evolution equations using Fredholm determinants. Electronic Research Archive, 2021, 29 (2) : 2141-2147. doi: 10.3934/era.2020109
[13]

Filippo Gazzola. On the moments of solutions to linear parabolic equations involving the biharmonic operator. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3583-3597. doi: 10.3934/dcds.2013.33.3583
[14]

Ai-Ling Yan, Gao-Yang Wang, Naihua Xiu. Robust solutions of split feasibility problem with uncertain linear operator. Journal of Industrial & Management Optimization, 2007, 3 (4) : 749-761. doi: 10.3934/jimo.2007.3.749
[15]

Phuong Le. Liouville theorems for stable weak solutions of elliptic problems involving Grushin operator. Communications on Pure & Applied Analysis, 2020, 19 (1) : 511-525. doi: 10.3934/cpaa.2020025
[16]

Pasquale Candito, Giovanni Molica Bisci. Multiple solutions for a Navier boundary value problem involving the $p$--biharmonic operator. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 741-751. doi: 10.3934/dcdss.2012.5.741
[17]

Caiping Liu, Heungwing Lee. Lagrange multiplier rules for approximate solutions in vector optimization. Journal of Industrial & Management Optimization, 2012, 8 (3) : 749-764. doi: 10.3934/jimo.2012.8.749
[18]

Leszek Gasiński, Nikolaos S. Papageorgiou. Periodic solutions for nonlinear nonmonotone evolution inclusions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 219-238. doi: 10.3934/dcdsb.2018015
[19]

Benoît Pausader, Walter A. Strauss. Analyticity of the nonlinear scattering operator. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 617-626. doi: 10.3934/dcds.2009.25.617
[20]

Vittorio Martino. On the characteristic curvature operator. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1911-1922. doi: 10.3934/cpaa.2012.11.1911

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences