Operator monotone functions of several variables

arXiv:math/0205147v1[math.OA]14May2002OperatormonotonefunctionsofseveralvariablesFrankHansenAbstractWeproposeanotionofoperatormonotonicityforfunctionsofseveralvari-ables,whichextendsthewellknownnotionofoperatormonotonicityforfunc-tionsofonlyonevariable.Thenotionischosensuchthatafundamentalrela-tionshipbetweenoperatorconvexityandoperatormonotonicityforfunctionsofonevariableisextendedalsotofunctionsofseveralvariables.1IntroductionandmainresultThenotionofoperatorconvexityforfunctionsofseveralvariableshasbeenextensivelystudiedintheliterature.Thefirststepistodefinethefunctionalcalculusforfunctionsofseveralvariables.Thiscanbedoneinthefollowingway:LetI1,...,Ikberealintervalsandletf:I1×···×Ik→RbeaBorelmeasurableandessentiallyboundedfunction.Letx=(x1,...,xk)beak-tupleofboundedself-adjointoperatorsonHilbertspacesH1,...,HksuchthatthespectrumofxiiscontainedinIifori=1,...,k.Wesaythatsuchak-tupleisinthedomainoff.Ifxi=ZIiλiEi(dλi)i=1,...,kisthespectraldecompositionofxi,wedefinef(x)=ZI1×···×Ikf(λ1,...,λk)E1(dλ1)⊗···⊗Ek(dλk)(1)asaboundedself-adjointoperatoronH1⊗···⊗Hk,cf.[4,1,9].IftheHilbertspacesareoffinitedimension,thentheaboveintegralsbecomefinitesums,andwemayconsiderthefunctionalcalculusforarbitraryrealfunctions.ThisconstructionextendsthedefinitionofKor´anyi[9]forfunctionsoftwovariablesandhavethepropertythatf(x1,...,xk)=f1(x1)⊗···⊗fk(xk),wheneverfcanbeseparatedasaproductf(t1,...,tk)=f1(t1)···fk(tk)ofkfunc-tionseachdependingononlyonevariable.Remark1.1Onemightconsiderthefunctionalcalculusonlyforcommutingopera-torsx1,...,xkonasingleHilbertspaceHanddefinefcom(x1,...,xk)=Zf(λ1,...,λk)dE(λ1,...,λk)11INTRODUCTIONANDMAINRESULT2asanoperatoronH,whereEistheproductmeasureofthecommutingspectralmeasuresassociatedwitheachoftheoperators.ThisapproachwassuggestedbyPedersenandLiebin[12,10].Ourdefinitioninequation(1)canthenbewrittenasf(x1,...,xk)=fcom(x1⊗1⊗···⊗1,...,1⊗···⊗1⊗xk)forarbitrarynon-commutingoperatorsx1,...,xkonH.Ifhowevertheoperatorsx1,...,xkdocommute,thenthereisaself-adjointprojectionPonH⊗···⊗HwithrangeisomorphictoHsuchthatfcom(x1,...,xk)=Pf(x1,...,xk)P.Thetwoapproachesarethusessentiallyequivalent.Oncethefunctionalcalculusisdefined,wesaythatafunctionf:I1×···×Ik→Risoperatorconvex,iffiscontinuousandtheoperatorinequalityf(λx+(1−λ)y)≤λf(x)+(1−λ)f(y)∀λ∈[0,1]holdsforallk-tuplesofself-adjointoperatorsx=(x1,...,xk)andy=(y1,...,yk)inthedomainoffactingonanyHilbertspacesH1,...,Hk.Thedefinitionismeaningfulsincealsothek-tupleλx+(1−λ)yisinthedomainoff.Wesaythatfismatrixconvexoforder(n1,...,nk),iftheoperatorinequalityholdsforoperatorsonHilbertspacesoffinitedimensions(n1,...,nk).Theaimofthispaperistodefinethenotionofanoperatormonotonefunctionalsoforfunctionsofseveralvariables.Thedefinitionshould,whenrestrictedtofunctionsofonlyonevariable,beasimplereformulationoftheordinaryconditionforsuchfunctions.Wealsowantthefollowingtheoremtobevalid.Theorem1.2Letf:[0,α1[×···×[0,αk[→Rbeacontinuousrealfunction.Thefollowingstatementsareequivalent:(i)fisoperatorconvex,andf(r1,...,rk)≤0ifri=0forsomei=1,...,k.(ii)Thefunctiong:]0,α1[×···×]0,αk[→Rdefinedbysettingg(r1,...,rk)=r−11···r−1kf(r1,...,rk)isoperatormonotone.Thetheoremaboveisknowntobevalidforfunctionsofonevariable[7,2.4Theorem],andtheextensiontofunctionsofseveralvariablesseemstobeverynatural.OurnotionofoperatormonotonicityforfunctionsofseveralvariablesisultimatelygiveninDefinition2.14,butitdependsonintermediarynotionsandresultsgiveninDefinition2.1,Definition2.2,Definition2.3,andCorollary2.13.Beforeproceedingwiththisprogramme,weshallbrieflydiscussotherpossibledefinitionsofoperatormonotonicityforfunctionsofseveralvariables,whichweulti-matelyhaverejected.Proposition1.3Letfbeanon-negativecontinuousfunctionofkvariablesdefinedinthefirstquadrant[0,∞[×···×[0,∞[.Iffismatrixconcaveoforder(n1,...,nk),then0≤xi≤yii=1,...,k⇒f(x)≤f(y)forarbitraryk-tuplesofpositivesemi-definitematricesx=(x1,...,xk)andy=(y1,...,yk)oforder(n1,...,nk).1INTRODUCTIONANDMAINRESULT3Proof:Lettheappropriatek-tuplesofmatricesbechosenandtakeλ∈[0,1[.Wesetzi=λ(1−λ)−1(yi−xi)andnoticethatλyi=λxi+(1−λ)ziandzi≥0fori=1,...,k.Sincefismatrixconcaveandnon-negativeweobtainf(λy)≥λf(x)+(1−λ)f(z)≥λf(x)wherez=(z1,...,zk).Theresultnowfollowsbylettingλtendtoone.QEDTheconverseisnottrue.Thefunctionoftwovariablesf(r1,r2)=r1r2isindeedmatrixincreasingofanyorderinthesensethatf(x1,x2)=x1⊗x2≤y1⊗y2=f(y1,y2)for0≤x1≤y1and0≤x2≤y2,butitisnotevenconcaveasarealfunction.How-ever,thesituationisquitedifferentforfunctionsofonlyonevariable.Mathias[11]showedthatafunction,definedonthepositiverealhalf-lineandmatrixmonotoneofordern,ismatrixconcaveoforder[n/2].Itfollowsfrom[3,7],althoughnotstatedexplicitely,thatafunction,definedontherealpositivehalf-lineandmatrixmonotoneoforder4n,ismatrixconcaveofordern.IfwerelaxMathias’resultonlyveryslightlyandaresatisfiedwithprovingthatafunctionf:[0,∞[→R,matrixmonotoneoforder2n,ismatrixconcaveofordern,thenthefollowingverysimpleargumentwilldo.Letx1,x2bepositivedefinitematricesofordernandnotice[3]thattoagivenε0theinequalityV∗x100x2V=12x1+x2x2−x1x2−x1x1+x2≤2−1(x1+x2)+ε00λisvalidf