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MaratV.Markin ElementaryOperatorTheory Also of Interest ElementaryFunctionalAnalysis MaratV.Markin,2018 ISBN978-3-11-061391-9,e-ISBN(PDF)978-3-11-061403-9, e-ISBN(EPUB)978-3-11-061409-1 RealAnalysis.MeasureandIntegration MaratV.Markin,2019 ISBN978-3-11-060097-1,e-ISBN(PDF)978-3-11-060099-5, e-ISBN(EPUB)978-3-11-059882-7 ComplexAnalysis.TheoryandApplications TeodorBulboacǎ,SantoshB.Joshi,PranayGoswami,2019 ISBN978-3-11-065782-1,e-ISBN(PDF)978-3-11-065786-9, e-ISBN(EPUB)978-3-11-065803-3 FunctionalAnalysiswithApplications SvetlinG.Georgiev,KhaledZennir,2019 ISBN978-3-11-065769-2,e-ISBN(PDF)978-3-11-065772-2, e-ISBN(EPUB)978-3-11-065804-0 AppliedNonlinearFunctionalAnalysis.AnIntroduction NikolaosS.Papageorgiou,PatrickWinkert,2018 ISBN978-3-11-051622-7,e-ISBN(PDF)978-3-11-053298-2, e-ISBN(EPUB)978-3-11-053183-1 Marat V. Markin Elementary Operator Theory | MathematicsSubjectClassification2010 47-01,47A10,47A30,47A35,47A56,47A60,47B07,47B25,46-01,46A30,46A35,46A45,46E15 Author Dr.MaratV.Markin CaliforniaStateUniversity,Fresno DepartmentofMathematics 5245N.BackerAvenue,M/SPB108 Fresno,California93740-8001 USA [email protected] ISBN978-3-11-060096-4 e-ISBN(PDF)978-3-11-060098-8 e-ISBN(EPUB)978-3-11-059888-9 DOIhttps://doi.org/10.1515/9783110600988 LibraryofCongressControlNumber:2019951807 BibliographicinformationpublishedbytheDeutscheNationalbibliothek TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie; detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de. ©2020WalterdeGruyterGmbH,Berlin/Boston Coverimage:Mordolff/GettyImages Typesetting:VTeXUAB,Lithuania Printingandbinding:CPIbooksLecks,GmbH www.degruyter.com | Tothebeautyandpowerofmathematics. Preface Mathematicsisthemostbeautifulandmostpowerfulcreationofthehumanspirit. StefanBanach AFewWordsontheSubjectofOperatorTheory AswasshrewdlyobservedbyS.KreinandYu.Petunin,“Tostudyaproblem,onemust chooseaspaceandstudythecorrespondingfunctionals,operators,etc.init....The choiceofthespaceinwhichtheproblemisstudiedispartlyconnectedwiththesub- jectiveaimssetbytheinvestigator.Apparently,theobjectivedataareonlytheopera- torsthatappearintheequationsoftheproblem.Onthisaccount,itseemstousthat theoriginalandbasicconceptoffunctionalanalysisisthatofanoperator”(see,e.g., [44]).Thus,inthecontextoftheancientchicken-eggargument,theconceptofspace emergesasconsequentialtothatofanoperator. Operatortheoryisamodern,vast,andrapidlydevelopingbranchoffunctional analysis,whichaddressesoperators,mostnotablylinear.Combiningprofoundlyab- stract nature with extensive applicability, which encompasses ordinary and partial differentialequations,integralequations,calculusofvariations,quantummechanics, andmuchmore,operatortheoryisapowerfulapparatusforsolvingdiverseproblems. Acourseinit,mostcertainly,deservestobeavitalpartofacontemporarygraduate mathematicscurriculum,increasingitsvaluenotonlyforgraduatestudentsmajoring inmathematicsbutalsoforthosemajoringinphysics,science,andengineering. Book’sPurposeandTargetedAudience Thebookisintendedasatextforaone-semesterMaster’slevelgraduatecourseinop- eratortheorytobetaughtwithintheexistingconstraintsofthestandardfortheUnited Statesgraduatecurriculum(fifteenweekswithtwoseventy-five-minutelecturesper week).Consideringtheabove,thisisanintroductorytextonthefundamentalsofop- eratortheorywithprerequisitesintentionallysetnothigh,thestudentsnotbeingas- sumedtohavetakengraduatecourseseitherinanalysis(realorcomplex)orgeneral topology,tomakethecourseaccessibleandattractivetoawideraudienceofSTEM (science,technology,engineering,andmathematics)graduatestudentsoradvanced undergraduateswithasolidbackgroundincalculusandlinearalgebra.Designedto teachaone-semesteroperatortheorycourse“fromscratch”,notasasequeltoafunc- tionalanalysiscourse,thisbookcannotbuthaveacertainnontrivialmaterialoverlap withtheauthor’srecenttextbook[45]remaining,however,verydistinctfromthelat- terbythescopeandthelearningoutcomes,withthebasicsofthespectraltheoryof linearoperatorstakingthecenterstage. https://doi.org/10.1515/9783110600988-202 VIII | Preface Book’sScopeandSpecifics Thebookconsistsofsixchaptersandanappendix,takingthereaderfromthefunda- mentalsofabstractspaces(metric,vector,normedvector,andinnerproduct),theBa- nachFixed-PointTheoremanditsapplications,suchasPicard’sExistenceandUnique- ness Theorem, through the basics of linear operators, two of the three fundamental principles(theUniformBoundednessPrincipleandtheOpenMappingTheoremandits equivalents:theInverseMappingandClosedGraphTheorems),totheelementsofthe spectraltheory,includingGelfand’sSpectralRadiusTheoremandtheSpectralTheo- remforCompactSelf-AdjointOperators,anditsapplications,suchasthecelebrated LyapunovStabilityTheoremandtheMeanErgodicityTheorem[48,Theorem4.1],the latterbeingaresultoftheauthor’sownresearch. ThecourseisdesignedtobetaughtstartingwithChapter2,Chapter1outlining certainnecessarypreliminariesandbeingreferredtowhenevertheneedarises. TheAppendixgivesaconcisetreatiseoftheAxiomofChoice,itsequivalents(the HausdorffMaximalPrinciple,Zorn’sLemma,andZermelo’sWell-OrderingPrinciple), andorderedsets,whichunderlieseveralfundamentalstatementsofthecourse,such astheBasisTheorem(Theorem3.2). Conceivedasatexttobeusedinaclassroom,thebookconstantlycallsforthestu- dent’sactivelymasteringtheknowledgeofthesubjectmatter.Thereareproblemsat theendofeachchapter,startingwithChapter2andtotaling150.Theseproblemsare indispensableforunderstandingthematerialandmovingforward.Manyimportant statements,suchastheFundamentalSequencewithConvergentSubsequenceProposi- tion(Proposition2.22,Section2.18,Problem25),aregivenasproblems;alotofthem arefrequentlyreferredtoandusedinthebook.Therearealso432Exercisesthrough- outthetext,includingChapter1andtheAppendix,whichrequireofthestudentto proveorverifyastatementoranexample,fillincertaindetailsinaproof,orprovide anintermediatesteporacounterexample.Theyarealsoaninherentpartofthema- terial.Moredifficultproblems,suchasSection4.7,Problem25,aremarkedwithan asterisk;manyproblemsandexercisesaresuppliedwith“existential”hints. ThebookisgenerousonExamplesandcontainsnumerousRemarksaccompany- ingeverydefinitionandvirtuallyeachstatementtodiscusscertainsubtleties,raise questions on whether the converse assertions are true, whenever appropriate, or whethertheconditionsareessential. Asamplydemonstratedbyexperience,studentstendtobetterrememberstate- mentsbytheirnamesratherthanbynumbers.Thus,adistinctivefeatureofthebook isthateverytheorem,proposition,corollary,andlemma,unlessalreadypossessinga name,isendowedwithadescriptiveone,makingiteasiertoremember,which,inthis author’shumbleopinion,isquiteabargainwhenthepriceforbetterunderstanding andretentionofthematerialisonlyalittleclumsinesswhilemakingalongerrefer- ence.Eachstatementisreferredtobyitsnameandnotjustthenumber,e.g.,theNorm EquivalenceTheorem(Theorem3.9),asopposedtomerelyTheorem3.9. Acknowledgments | IX Withnopretenseonfurnishingthehistoryofthesubject,thetextprovidescertain datesandlistseveryrelatednameasafootnote. Acknowledgments Firstandforemost,Iwishtoexpressmyheartfeltgratitudetomymother,Svetlana A.Markina,forherunfailingloveandsupport,withoutwhichthisandmanyother endeavorsofminewouldhavebeenimpossible. My utmost appreciation goes to Mr. Edward Sichel, my pupil and graduate ad- visee,forhisinvaluableassistancewithproofreadingandimprovingthemanuscript. IamverythankfultoDr.PrzemyslawKajetanowicz(DepartmentofMathematics, CSU,Fresno)forhiskindaidwithgraphics. My sincere acknowledgments are also due to Dr. Apostolos Damialis, formerly WalterdeGruyterGmbH AcquisitionsEditorinMathematics,forseeingvalueinmy manuscript and making authors his highest priority, Ms. Nadja Schedensack, Wal- terdeGruyterGmbHProjectEditorinMathematicsandPhysics,forsuperbefficiency inmanagingallprojectrelatedmatters,aswellasMs.InaTalandienėandMs.Ieva Spudulytė,VTeXBookProduction,fortheirexperteditorialandLATEXtypesettingcon- tributions. Clovis,California,USA MaratV.Markin June–July2019