Spectral Theory and Applications of Linear Operators and Block Operator Matrices PDF
Preview Spectral Theory and Applications of Linear Operators and Block Operator Matrices
Aref Jeribi Spectral Theory and Applications of Linear Operators and Block Operator Matrices Spectral Theory and Applications of Linear Operators and Block Operator Matrices Aref Jeribi Spectral Theory and Applications of Linear Operators and Block Operator Matrices 123 ArefJeribi DepartmentofMathematics UniversityofSfax Sfax,Tunisia ISBN978-3-319-17565-2 ISBN978-3-319-17566-9 (eBook) DOI10.1007/978-3-319-17566-9 LibraryofCongressControlNumber:2015102923 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper SpringerInternationalPublishingAGSwitzerlandispartofSpringerScience+BusinessMedia(www. springer.com) To mymotherSania, myfatherAli, mywifeFadoua, mychildrenAdamandRahma, mybrothersSofienandMohamedAmin, mysisterElhem, mymother-in-law Zineb, myfather-in-law Ridha,and allmembersofmyextendedfamily Preface Several books have been devoted to the spectral theory and its applications. However, this volume is very special as compared with the previous ones. For example, the perturbation approach has been, for a long time, extensively studied and considered as one of the most useful methods used in order to study some mathematicalandappliedproblems. The main idea is that, if we know something about the solution for an easier problem,lying“close”totheonewearestudying,thenwecansaysomethingabout ourproblem,providedthatthedifferenceortheperturbationissufficientlyweak. In view of more advanced applications, especially the ones dealing with com- plicated evolutional problems in Physics, Chemistry, Technology, Biology, etc., where the natural setting doesn’t involve single operators but operator matrices andpolynomialoperatorpencils,theconceptofcompactperturbationsisveryoften used,anditwasshownthattheywerenotsufficientforhandlingsuchproblems.The mainadvantageofthisbookisthedetaileddescriptionofthewaysshowinghowthe compactness condition can be relaxed, in a very general Banach space setting, so thatthepreviouslyimpossibleproblemsbecomesuddenlysolvable.Themethodof extendingresultsisnotunique.Thatiswhywehavetodevotealotofspaceinorder todescribethedifferentextensionsoftheclassicalnotions,andtodemonstratehow theyspecificallyworkindifferentapplications. More precisely, it is well known that the essential spectrum of an operator A consistsofthosepointsofthespectrumwhichcannotberemovedfromthespectrum bytheadditiontoAofacompactoperator.Themostpowerfulresultobtainedinmy thesisisthat,inL -spaces,theessentialspectrumofanoperatorAisnothingelse 1 but the largest subset of the spectrum of A which remains invariant under weakly compact perturbations ofA.Thisunexpected resulthasopened many prospectsto developinnovativewaysleadingtoarigorousstudyoftheFredholmtheoryandin the whole book, we give an account of the recent research on the spectral theory by presenting a wide panorama of techniques including the weak topology, which vii viii Preface contributestoanextrainsighttotheclassicalresultsandenablesustosolveconcrete problemsfromtransporttheoryarisingintheirnaturalsetting(L -spaces).Themain 1 topicsinclude: • Riesztheoryofpolynomiallycompactoperators. • TimebehaviorofsolutionsforanabstractCauchyproblemonBanachspaces. • Fredholm theory and characterization of essentialspectra by means of measure of noncompactness, demicompact operator, measure of weak noncompactness, andgraphmeasures. • S-essentialspectraandessentialpseudospectra. • Spectraltheoryofblockoperatormatrices. • Spectralgraphtheory. • Applicationsinmathematicalphysicsandbiology. Wedohopethatthisbookwillbeveryusefulforresearchers,sinceitrepresents notonlyacollectionofapreviouslyheterogeneousmaterial,butalsoaninnovation throughseveralextensions. Of course, it is impossible for a single book to cover such a huge field of research. In making personal choices for inclusion of material, we tried to give useful complementary references in this research area, hence probably neglecting some relevant works. We would be very grateful to receive any comments from readers and researchers, providing us with some information concerning some missingreferences. We would like to thank Salma Charfi for the improvement she has made in the introduction of this book. So, we are indebted to her. We would like to thank NedraMoallafortheimprovementsshehasmadeconcerningthespectralmapping theorem. We would also like to thank Aymen Ammar for the improvements he has made throughout this book. So, we are very grateful to him. Concerning the chapter related to graph theory, we were fortunate to have the help of Hatem Baloudi, who assisted in the preparation of this chapter. So, we are indebted to him.WewouldliketothankProfessorSylvainGoléniaforhisgenerouspermission to integrate, in this book, the results of Hatem Baloudi dealing with the graph theory.Moreover,wewouldliketomentionthatthethesisworkresults,performed undermydirection,bymyformerstudentsandpresentlycolleaguesNedraMoalla, Afif Ben Amar, Faiçal Abdmouleh, Boulbeba Abdelmoumen, Salma Charfi, Ines Walha,BilelKrichen,OmarJedidi,SoniaYengui,AymenAmmar,NaouelBenAli, RihabMoalla,HatemBaloudi,MohammedZeraiDhahri,andBilelBoukettaya,the obtainedresultshavehelpedusinwritingthisbook.Lastbutnotleast,wewouldlike tothankRidhaDamakforimprovingtheEnglishofallchaptersofthisbook.Finally, we apologize in case we have forgotten to quote any author who has contributed, directlyorindirectly,tothiswork. Sfax,Tunisia ArefJeribi June2015 Contents 1 Introduction ................................................................ 1 1.1 SpectralTheoryandCauchyProblem ............................. 2 1.2 TimeBehaviorofSolutionstoanAbstractCauchy ProblemonBanachSpaces......................................... 4 1.3 FredholmTheoryandEssentialSpectra........................... 8 1.4 S-EssentialSpectraandEssentialPseudospectra ................. 13 1.5 SpectralTheoryofBlockOperatorMatrices...................... 15 1.6 SpectralGraphTheory.............................................. 16 1.7 ApplicationsinMathematicalPhysicsandBiology............... 18 1.8 OutlineofContents................................................. 20 2 Fundamentals............................................................... 23 2.1 BasicProperties..................................................... 23 2.1.1 ClosedandClosableOperators.......................... 23 2.1.2 AdjointOperator......................................... 24 2.1.3 ElementaryResults ...................................... 25 2.1.4 FredholmOperators...................................... 26 2.1.5 Spectrum ................................................. 28 2.1.6 RelativelyBoundednessandRelatively Compactness............................................. 29 2.1.7 SumofClosedOperators................................ 31 2.1.8 StrictlySingularandStrictlyCosingularOperators.... 32 2.1.9 FredholmandSemi-FredholmPerturbations........... 33 2.1.10 Dunford–PettisProperty................................. 35 2.2 BasicNotions ....................................................... 36 2.2.1 BasicsonBoundedFredholmOperators ............... 37 2.2.2 GapTopology............................................ 42 2.2.3 Semi-RegularandEssentiallySemi-Regular Operators................................................. 44 2.2.4 BasicsonUnboundedFredholmOperators............. 48 ix
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