Table Of ContentGrundlehren 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
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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)
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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
