Loewner's Theorem on Monotone Matrix Functions PDF
Preview Loewner's Theorem on Monotone Matrix Functions
Grundlehren der mathematischen Wissenschaften 354 A Series of Comprehensive Studies in Mathematics Barry Simon Loewner’s Theorem on Monotone Matrix Functions Grundlehren der mathematischen Wissenschaften A Series of Comprehensive Studies in Mathematics Volume 354 Editors-in-Chief AlainChenciner,IMCCE-ObservatoiredeParis,Paris,France JohnCoates,EmmanuelCollege,Cambridge,UK S.R.S.Varadhan,CourantInstituteofMathematicalSciences,NewYork,NY,USA SeriesEditors PierredelaHarpe,UniversitédeGenève,Genève,Switzerland NigelJ.Hitchin,UniversityofOxford,Oxford,UK AnttiKupiainen,UniversityofHelsinki,Helsinki,Finland GillesLebeau,UniversitédeNiceSophia-Antipolis,Nice,France Fang-HuaLin,NewYorkUniversity,NewYork,NY,USA ShigefumiMori,KyotoUniversity,Kyoto,Japan BaoChauNgô,UniversityofChicago,Chicago,IL,USA DenisSerre,UMPA,ÉcoleNormaleSupérieuredeLyon,Lyon,France NeilJ.A.Sloane,OEISFoundation,HighlandPark,NJ,USA AnatolyVershik,RussianAcademyofSciences,St.Petersburg,Russia MichelWaldschmidt,UniversitéPierreetMarieCurieParis,Paris,France Grundlehren der mathematischen Wissenschaften (subtitled Comprehensive StudiesinMathematics),Springer’sfirstseriesinhighermathematics,wasfounded by Richard Courant in 1920. 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Moreinformationaboutthisseriesathttp://www.springer.com/series/138 Barry Simon Loewner’s Theorem on Monotone Matrix Functions 123 BarrySimon DivisionofPhysics,Math,andAstronomy Caltech,Pasadena,CA,USA ISSN0072-7830 ISSN2196-9701 (electronic) GrundlehrendermathematischenWissenschaften ISBN978-3-030-22421-9 ISBN978-3-030-22422-6 (eBook) https://doi.org/10.1007/978-3-030-22422-6 MathematicsSubjectClassification:26A48,26A51,47A56,47A63 ©SpringerNatureSwitzerlandAG2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface ThisbookisalovepoemtoLoewner’stheorem.Thereareothermathematicallove poems,althoughnotmany.Onesign,notalwayspresentandnotfoolproof,isthat like this book, the author has included pictures of some of the main figures in the development of the subject under discussion. The telltale sign is that the reader’s initial reaction is “how can there a whole book on that subject” (although not all narrowbooksarelovepoems). Loewner’s theorem concerns the theory of monotone matrix functions, i.e. functions, f so that A ≤ B ⇒ f(A) ≤ f(B) for pairs of selfadjoint matrices. Thatthisisasubtlenotionisseenbythefact(seeCorollary14.3)thatiff :R→R is monotone on all pairs of 2×2 matrices, then f is affine! So Loewner realized one needed to fix a proper interval (a,b) ⊂ R, demand that f : (a,b) → R, and onlydemandthemonotonicityresultforpairsAandB allofwhoseeigenvaluesare in(a,b).In1934,CharlesLoewnerprovedtheremarkableresultthatf on(a,b)is matrixmonotoneonallsuchn×npairs(foralln)ifandonlyiff isrealanalytic on (a,b) and has an analytic continuation to the upper half plane with a positive imaginarypartthere.Thatfunctionswiththispropertyarematrixmonotonefollows inafewlinesfromtheHerglotzrepresentationtheorem,soIcallthathalftheeasy half.Theotherdirectionisthehardhalf.ManyapplicationsofLoewner’snotionof matrixmonotonefunctionsinvolveexplicitexamplesandsoonlytheeasyhalf. Matrix monotonicity is an algebraic statement, but Loewner’s theorem says it is equivalent to an analytic fact. One fascination of the subject is the mix of the algebraicandtheanalyticinitsstudy. Interestinglyenough,thisisnotthefirstlovepoemtoLoewner’stheorem.Forty- five years ago, in 1974, Springer published W. F. Donoghue’s Monotone Matrix FunctionsandAnalyticContinuation[81]inthesameserieswhereIampublishing thisbook.Donoghueexplainedthat,atthetimeofwriting,therewerethreeexisting proofsof(thehardhalfof)Loewner’stheorem,allverydifferent:Loewner’soriginal proof, the proof of Bendat–Sherman, and the proof of Korányi. In fact, as we’ll discuss several times in this book, there was a fourth proof which workers in the field didn’t seem to realize was there—it was due to Wigner–von Neumann and appearedintheAnnalsofMathematics!ThemaingoalofDonoghue’sbookwasto v vi Preface exposethosethreeproofswiththeirunderlyingmathematicalbackground.Healso hadseveralnewresultsincludingconversesoftwotheoremsofLoewner’sstudent Otto Dobsch. Donoghue also discussed applications to the theory of schlicht and Herglotzfunctions. Similarly,themaingoalofthecurrentbookistoexpose many oftheproofsof the hard half that now exist. Indeed, we will give 11 proofs in all. As I’ll explain later, there is a 12th proof which I don’t expose. Of course, it is not always clear when two proofs are really different and when one proof should be regarded as a variantofanother.Iampreparedtodefendthenotionthattheseproofsaredifferent (although there are relations among them) and that a proof like Hansen’s in [133] shouldbeviewedasavariant(albeitaninterestingvariant)ofHansen’searlierproof withPedersen.ButI’dagreesomemightdiffer. In any event, it is striking that there are no really short proofs and that the underlying structure of the proofs is so different. I like to joke that it is almost as if every important result in analysis is the basis of some proof of Loewner’s theorem. To mention a few of the underlying machines that lead to proofs: the moment problem, Pick’s theorem, commutant lifting theorems, the Krein–Milman theorem,andMellintransforms. Shortly after Loewner’s 1934 paper, his student Fritz Kraus defined and began thestudyofmatrixconvexfunctions.DonoghueincludesKraus’1936paperinhis bibliographybutneverreferstoitinthetextanddoesn’tdiscussthesubjectofmatrix convex or concave functions. In the years since his book, it has become clear that thetwosubjectsareintertwinedindeepways.Somybookalsohasalotofmaterial onmatrixconvexfunctions. We let M (a,b) be the functions monotone on n×n matrices. We’ll see that n Mn+1 is a strictly proper subset of Mn. Loewner’s theorem gives an effective description of M∞ ≡ ∩Mn, and we’ll see in Chapter 14 that there is an effective description of M but there is no especially direct description of M . One other 2 n subjectweexploreinvolvespropertiesofM anditsconvexanalogforfixedfinite n n. Ishouldsaysomethingaboutthehistoryofthisbook.IfirstlearnedofLoewner’s theorem in graduate school 50 years ago, probably from Ed Nelson, one of my mentors. I learned the proof of the easy half and a variant of it appears in Reed– Simon. The easy half seemed more useful since it told one certain function was matrixmonotoneandthatwaswhatcouldbeapplied.About2000,Ibecamecurious how one proved the hard half and looked at the expositions in Donoghue and Bhatia’sbooks.IwasintriguedthatbothquotedapaperofWigner–vonNeumann claiming it had applications to quantum physics. I looked up the paper and was surprisedtodiscoverthattherewasnoapplicationtoquantumphysicsbuttherewas acomplete anddifferentproofofLoewner’s theorem,whichLoewner,Donoghue, andBhatiadidn’tseemtorealizewasthere!Forseveralyears,Igaveamathematics colloquiumcalled“TheLostProofofLoewner’sTheorem.”Ifoundmynewproof that appears in Chapter 20 which is, in many ways, the most direct proof. After one of the colloquia, around 2005, I learned of Boutet de Monvel’s unpublished proofandstartedonthisbook.Ifinishedabouthalfandthenputitasidein2007to Preface vii focusonmyfive-volumeComprehensiveCourse[325–329].Aftercompletingthat, Ireturnedtothisbookandcompletedit.TheearlypartofthebookwastypedinTEX bymywonderfulsecretary,CherieGalvez,whowelosttocancerbeforeIreturned totheproject.Ioftenmissher. There is some literature on multivariable extensions of Loewner’s theorem. The earliest such papers are a series (Korányi [187], Singh–Vasudeva [315], and Hansen [131]) that considers operators on distinct spaces, A on H , and defines j j f(A ,...,A ) on the tensor product H ⊗···⊗H . Recently, there have been 1 n 1 n several papers (Agler et al. [3], Najafi [230], Pascoe and Tully-Doyle [259], and Pálfia [256]) that involve functions of several variables, even non-commuting variables,onasingleHilbertspace.Thesepapersarecomplicated,andthesubject is in flux, so it seemed wisest not to discuss the subject in detail here. However, we should mention that Pálfia [256] says that his multivariable proof specialized toonevariableprovidesanewproofoftheclassicalresult,the12thproofreferred toabove.Incorrespondence,hetoldmethattherestrictiontoonevariabledoesn’t especially simplify his proof (his preprint is 40 pages). He didn’t believe it could be shortened to less than about 25 pages and that doesn’t count any mathematical backgroundonthetoolsheuses,soIdecidednottotrytoincludeit. The11proofsappearinPartIIofthisbookwhichalsohasbackgroundontools like Pick’s theorem and rational approximation. There is also a discussion of the analog of Loewner’s theorem when (a,b) is replaced by a more general open set. PartIsetsthebackgroundforthetheoremandalsoforthetheoryofmatrixconvex functions.PartIIIdiscussessomeapplications,manyduetoAndo.Eachpartbegins withabriefintroductionthatsummarizesthecontentandcontextofthatpart. KarelLöwnerwasbornnearPraguein1893andprovedhisgreattheoremwhile a professor at the Charles University in Prague. In 1938, he fled from there to the UnitedStates(hewasarrestedwhentheNazisenteredPragueprobablybecauseof his left-wing political activities rather than the fact that he was Jewish) where he made a decision to change his name to the less German Charles Loewner. I have decided to respect his choice by calling the result Loewner’s theorem. The reader shouldbewarnedthatmuchoftheliteraturereferstoLöwner’stheorem.Ofcourse, we use the original spelling in the bibliography when that is what appeared in the author’snameorinthepublishedtitleofanarticleorbook. We warn the reader that in our complex inner product, we use a universal conventioninthephysicscommunity,buttheminorityoneamongmathematicians namely,ourinnerproducts,arelinearinthesecondvectorandantilinearinthefirst. OthersymbolsarelistedinAppendixCjustbeforethebibliography.Wealsowarn the reader that when speaking of matrices, M, we will call them “positive” if and onlyif(cid:10)ϕ,Mϕ(cid:11) ≥ 0forallϕ anduse“strictlypositive”whenthatinnerproductis strictlypositiveforallnon-zeroϕ. Two individuals should be especially thanked for their aid in improving this book.MitsuruUchiyamasentmeextensivecorrectionstoanearlydraftofthisbook. Detailedcorrectionsofthekindhesentarelikegoldtoanauthor.OtteHeinävaara was kind enough to share his many insights into the subject of this book. It is a pleasuretoalsothankAnneBoutetdeMonvel,LarryBrown,FritzGesztesy,Frank viii Preface Hansen,IraHerbst,BillHelton,ElliottLieb,JohnMcCarthy,JamesPascoe,James Rovnyak, and Beth Ruskai for their useful discussions and/or correspondence in connectionwiththisbook. And,asalways,thankstomywife,Martha,forherloveandsupport. LosAngeles,CA,USA BarrySimon March2019 Contents PartI Tools 1 Introduction:TheStatementofLoewner’sTheorem................... 3 2 SomeGeneralities........................................................... 11 3 TheHerglotzRepresentationTheoremsandtheEasyDirection ofLoewner’sTheorem ..................................................... 17 4 MonotonicityoftheSquareRoot ......................................... 33 5 LoewnerMatrices .......................................................... 43 6 Heinävaara’sIntegralFormulaandtheDobsch–DonoghueTheorem 73 7 Mn+1 (cid:2)=Mn ................................................................. 85 8 Heinävaara’sSecondProofoftheDobsch–DonoghueTheorem ...... 89 9 Convexity,I:TheTheoremofBendat–Kraus–Sherman–Uchiyama.. 95 10 Convexity,II:ConcavityandMonotonicity ............................. 107 11 Convexity,III:Hansen–Jensen–Pedersen(HJP)Inequality........... 117 12 Convexity,IV:Bhatia–Hiai–Sano(BHS)Theorem ..................... 123 13 Convexity,V:StronglyOperatorConvexFunctions.................... 135 14 2×2Matrices:TheDonoghueandHansen–TomiyamaTheorems... 145 15 QuadraticInterpolation:TheFoias¸–LionsTheorem................... 153 PartII ProofsoftheHardDirection 16 PickInterpolation,I:TheBasics.......................................... 165 17 PickInterpolation,II:HilbertSpaceProof.............................. 175 18 PickInterpolation,III:ContinuedFractionProof...................... 183 ix
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