Non-Archimedean Operator Theory PDF
Preview Non-Archimedean Operator Theory
SPRINGER BRIEFS IN MATHEMATICS Toka Diagana François Ramaroson Non-Archimedean Operator Theory 123 SpringerBriefs in Mathematics SeriesEditors NicolaBellomo MicheleBenzi PalleE.T.Jorgensen TatsienLi RoderickMelnik OtmarScherzer BenjaminSteinberg LotharReichel YuriTschinkel G.GeorgeYin PingZhang SpringerBriefs in Mathematics showcases expositions in all areas of mathematics and applied mathematics. Manuscripts presenting new results or a single new result in a classical field, new field, or an emerging topic, applications, or bridges between new results and already published works, are encouraged. The series is intended for mathematicians and applied mathematicians. Moreinformationaboutthisseriesathttp://www.springer.com/series/10030 Toka Diagana • François Ramaroson Non-Archimedean Operator Theory 123 TokaDiagana FrançoisRamaroson DepartmentofMathematics DepartmentofMathematics HowardUniversity HowardUniversity Washington,DC,USA Washington,DC,USA ISSN2191-8198 ISSN2191-8201 (electronic) SpringerBriefsinMathematics ISBN978-3-319-27322-8 ISBN978-3-319-27323-5 (eBook) DOI10.1007/978-3-319-27323-5 LibraryofCongressControlNumber:2015957041 MathematicsSubjectClassification(2010):12J25,26E30,32P05,37P20,46S10,47S10,47A53,35P05 SpringerChamHeidelbergNewYorkDordrechtLondon ©TheAuthor(s)2016 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) Thisbookisdedicated,byTokaDiagana, to hiswifeSally,forherunyielding love, support,andexceptionalpatience. Thisbookisdedicated,byFrançois Ramaroson,tohiswifeLuna,forher unyieldinglove,support,andexceptional patience. Preface Thisbookisfocusedonthetheoryoflinearoperatorsonnon-ArchimedeanBanach spaces.Itistosomeextentasequeloftheauthors’recentworkonlinearoperators onnon-ArchimedeanBanachspacesaswellastheirspectraltheory.Therearenot manybooksexclusivelydealingwithoperatortheoryonnon-ArchimedeanBanach spaces or their variants, and the authors wish to add to the scarce literature on the subject. A minimum necessary background material has been gathered which will allowarelativelyfriendlyaccesstothebook. Beginning graduate students who wish to enter the field of non-Archimedean functional analysis should benefit from the material covered, but an expert reader mightalsofindsomeoftheresultsinterestingenoughtobesourcesofinspiration. Prerequisites for the book are the basic courses in classical real and complex analysis and some knowledge of basic functional analysis. Further, knowledge of basic algebra (groups, rings, fields, vector spaces) and some familiarity with p- adicnumbers,suchasGouvêa’sintroductorybookp-adicnumbers:AnIntroduction (secondedition,Springer,2003),willbeahugeplus.Thestudentwouldgainalong way knowing the first four chapters of the book Local Fields by Cassels (1986, LondonMathematicalSociety,StudentTexts3). Afterreadingthisbook,thereadermightbenefitagreatdealifhe/shemoveson forwardwithdeepermateriallikethebookSpectralTheoryandAnalyticGeometry over non-Archimedean Fields by Berkovich which is geared toward a systematic treatment of the spectral theory for Banach algebras (respectively, Rigid analytic geometry) or the books Non-Archimedean Functional Analysis by Schneider and Non-Archimedean Functional Analysis by van Rooij. Those readers with more arithmeticalinclinationswillprofitfromthebookp-adicAnalysis:AShortCourse onRecentWorkbyKoblitz. The topics treated in the book range from a basic introduction to non- Archimedean valued fields, free Banach spaces, and (possibly unbounded) linear operators in the non-Archimedean setting to the spectral theory for some classes of linear operators and their perturbations. Although some parts of the material are taken from the book Non-Archimedean Linear Operators and Applications by Diagana, this book is more comprehensive as it covers many new topics. It vii viii Preface emphasizes the role of the theory of Fredholm operators which is used as an importanttool.Thisapproachinthestudyofthespectraltheoryoflinearoperators shouldplaymorerolesinlargercontext thantheones covered inthebook,andin thisregard,thebookisagoodintroductiontothespectraltheoryoflinearoperators inthenon-Archimedeansetting.Explicitdescriptionsofthespectraofsomelinear operatorsareworkedout. Chapter1isofabackgroundnature.Itcoversnon-Archimedeanvaluedfieldsand containsmanydetailsandexamplesonnon-Archimedeanvaluations,thetopology inducedbythesevaluations,andtheirextensions.Sphericalcompletenessisdefined and some related properties are proved and illustrated with examples. The Krull valuationisintroduced. Chapter 2 is also of a background nature and covers non-Archimedean Banach spaces. These spaces are complete normed vector spaces over a complete non- Archimedeanvaluedfield.OfspecialinterestarethefreeBanachspaces,especially thep-adicHilbertspace,andtheyarestudiedindetail.Astructuretheoremforthe p-adicHilbertspaceisproved. Chapter 3 is on the bounded linear operators. Various properties are stated andproved.Finiterankoperators,completelycontinuousoperators,andFredholm operatorsarealldiscussedwithaviewtowardtheapplicationsinspectraltheory. Chapter4introducesandstudiespropertiesoftheShnirel’manintegral.Among other things, such an integral is used to construct the so-called Vishik spectral theorem. Chapter 5 contains the determination of the spectrum of a perturbation of a bounded diagonal operator by finite rank operators. The technique uses the theory ofFredholmoperators. Chapter6treatsgeneralunboundedoperators,closedoperators,andthespectrum ofunboundedoperatorsandtheunboundedFredholmoperators. Chapter 7 is devoted to the study of spectral theory for the perturbations of an unbounded operator by operators of finite rank or by completely continuous operators. Special emphasis is put on the computation of the essential spectrum oftheseperturbedunboundedlinearoperators. This book is intended for graduate and postgraduate students, mathematicians, and nonmathematicians such as physicists and engineers who are interested in functional analysis in the non-Archimedean context. Further, it can be used as an introductiontothestudyoflinearoperatorsingeneralandtothestudyofspectral theoryinotherspecialcases. Washington,D.C,USA TokaDiagana May,2015 FrançoisRamaroson Acknowledgments Theauthorswouldliketoexpresstheirthankstocolleagues,students,andcollabo- ratorswhohavehelpedinthecourseofthiswork. ix
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