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Journal
of
Functional
Analysis
www.elsevier.com/locate/jfa
Nevanlinna
representations
in
several
variables
✩J. Aglera,R. Tully-Doyleb, N.J. Youngc,d,∗
a Department of Mathematics, University of California at San Diego, CA 92103, USA
b Department of Mathematics, Hampton University, Hampton, VA 23668, USA cSchool of Mathematics, Leeds University, Leeds LS2 9JT,
England, United Kingdom
d School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne NE3 4LR, England, United Kingdom
a r t i c l e i n f o a bs t r a c t
Article history:
Received 28 December 2014 Accepted 2 February 2016 Available online 23 February 2016 Communicated by G. Schechtman
MSC: 32A30 30E20 30E05 47B25 47A10
Keywords: Pick class Cauchy transform Self-adjoint operator Resolvent
We generalize to several variables the classical theorem of NevanlinnathatcharacterizestheCauchytransformsof posi-tivemeasuresontherealline.WeshowthatfortheLoewner class,a largeclassofanalyticfunctionsthathavenon-negative imaginary part on theupper polyhalf-plane, there are rep-resentation formulae intermsof densely-definedself-adjoint operators on a Hilbertspace. We find four different repre-sentation formulae andweshow that every functionin the Loewnerclasshasoneofthefourrepresentations, correspond-ingpreciselytofourdifferentgrowthconditionsatinfinity.
© 2016TheAuthors.PublishedbyElsevierInc.Thisisan openaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).
✩ The first author was partially supported by the National Science Foundation Grant on Extending
Hilbert Space Operators DMS 1068830 and 1361720. The third author was partially supported by the UK Engineering and Physical Sciences Research Council grants EP/J004545/1 and EP/N03242X/1.
* Corresponding author at: School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne NE3 4LR, England, United Kingdom.
E-mail addresses:[email protected](R. Tully-Doyle), [email protected]
(N.J. Young).
http://dx.doi.org/10.1016/j.jfa.2016.02.004
1. Introduction
Inaclassicpaper [18]of 1922R. Nevanlinnasolved theproblem ofthedeterminacy ofsolutionsoftheStieltjesmomentproblem.Enrouteheprovedseveralothertheorems thathavesincebeeninfluential;inparticular,thefollowingtheorem,whichcharacterizes the Cauchytransforms of positivefinite measures μon R, has had aprofound impact onthedevelopmentof modernanalysis.LetP denotethePickclass, thatis, theset of analyticfunctionsontheupperhalf-plane,
Πdef= {z∈C: Imz >0},
thathavenon-negativeimaginary partonΠ.
Theorem 1.1 (Nevanlinna’s Representation). Let h be a function defined on Π. There existsafinite positivemeasureμon Rsuchthat
h(z) =
dμ
t−z (1.1)
ifand onlyif h∈ P and
lim inf
y→∞ y|h(iy)|<∞. (1.2)
Acloselyrelatedtheorem,alsoreferredtointheliteratureasNevanlinna’s Represen-tation,providesanintegralrepresentationforageneralelement ofP.
Theorem1.2.Afunctionh: Π→CbelongstothePickclassP ifandonlyifthereexist
a∈R,b≥0andafinite positiveBorelmeasureμ onRsuchthat
h(z) =a+bz+
1 +tz
t−z dμ(t) (1.3)
forallz∈Π.Moreover,foranyh∈ P,thenumbersa∈R,b≥0andthemeasureμ≥0
intherepresentation (1.3)are uniquelydetermined.
Whataretheseveral-variableanalogsofNevanlinna’stheorems?Inthispaperweshall proposefour typesofNevanlinnarepresentationforvarious subclassesofthen-variable PickclassPn,wherePn isdefinedto bethesetofanalyticfunctionshonthe
polyhalf-plane Πn such that Imh ≥ 0. In addition, we shall present necessary and sufficient
conditions for afunction defined on Πn to possess a representation of agiven typein
termsofasymptoticgrowthconditionsat∞.
h(z) =(A−z)−11,1L2(μ),
where A is the operation of multiplication by the independent variable on L2(μ) and
1 is the constantfunction 1.We proposethat anappropriate n-variable analog of the Cauchytransformistheformula
h(z1, . . . , zn) =
(A−z1Y1− · · · −znYn)−1v, v
H forz1, . . . , zn∈Π, (1.4)
where HisaHilbertspace,Aisadenselydefinedself-adjointoperatoronH,Y1,. . . ,Yn
are positivecontractionsonHsumming to1 andvis avectorinH.
Theorem 1.6 belowcharacterizesthosefunctionsonΠn thathavearepresentationof
theform(1.4).Tostatethistheoremwerequireanotionbasedonthefollowingclassical resultofPick[20].
Theorem1.3.AfunctionhdefinedonΠbelongstoP ifandonlyifthefunctionAdefined on Π×Πby
A(z, w) = h(z)−h(w)
z−w¯
is positivesemidefinite,that is, foralln≥1,z1,. . . ,zn∈Π,c1,. . . ,cn∈C,
A(zj, zi)cicj≥0.
Thefollowingtheorem,provedin[2],leadstoageneralizationofTheorem 1.3totwo variables.TheSchurclassofthepolydisc,denotedbySn,isthesetofanalyticfunctions
onthepolydisc Dn thatareboundedby1 inmodulus.
Theorem1.4.A functionϕdefinedonD2 belongstoS2 ifandonlyifthereexistpositive
semidefinite functionsA1 andA2 onD2×D2 suchthat
1−ϕ(μ)ϕ(λ) = (1−μ1λ1)A1(λ, μ) + (1−μ2λ2)A2(λ, μ). (1.5)
Bywayofthetransformations
z=i1 +λ
1−λ, λ= z−i
z+i, (1.6)
and
h(z) =i1 +ϕ(λ)
1−ϕ(λ), ϕ(λ) =
h(z)−i
h(z) +i, (1.7)
Theorem1.5.AfunctionhdefinedonΠ2 belongstoP2 ifandonlyif thereexistpositive
semidefinitefunctionsA1 andA2 onΠ2×Π2 suchthat
h(z)−h(w) = (z1−w1)A1(z, w) + (z2−w2)A2(z, w).
In the lightof Theorems 1.3 and 1.5 we define the Loewner class Ln to be the set
ofanalyticfunctionshonΠn withtheproperty thatthere exist npositivesemidefinite functionsA1,. . . ,An onΠn suchthat
h(z)−h(w) =
n
j=1
(zj−wj)Aj(z, w) (1.8)
for all z,w ∈ Πn. The Loewner class L
n played a key role in [4], which gave a
generalization to several variables of Loewner’s characterization of the one-variable operator-monotone functions [17]. As the following theorem makes clear, Ln also has
afundamentalroletoplayintheunderstandingofNevanlinnarepresentationsinseveral variables.
Theorem1.6. A functionhdefined on Πn has arepresentation of theform (1.4)if and
onlyif h∈ Ln and
lim inf
y→∞ y|h(iy, . . . , iy)|<∞. (1.9)
In the cases when n = 1 and n = 2, Theorems 1.3 and 1.5 assert that Ln = Pn,
andso for n= 1,Theorem 1.6is Nevanlinna’s classicalTheorem 1.1,and when n= 2,
Theorem 1.6 is a straightforward generalization of that result to two variables. When therearemorethantwovariables,itisknownthattheLoewnerclassisapropersubsetof thePickclass,Ln=Pn [19,22].Nevertheless,Nevanlinna’s resultsurvives asatheorem
about the representation of elements of Ln. Other than the work in [11] very little is
knownabouttherepresentationoffunctionsinPn forthreeor morevariables.
For a function h on Πn, we call the formula (1.4) a Nevanlinna representation of
type 1. Thus, Theorem 1.6 can be rephrased as the assertionthat hhas aNevanlinna representationoftype 1 ifand onlyifh∈ Ln andhsatisfiescondition(1.9).Somewhat
more complicated representation formulae are needed to generalize Theorem 1.2. We identify three further representation formulae, of increasing generality, and show that everyfunctioninLn hasarepresentation ofoneor moreofthefourtypes.
ForafunctionhdefinedonΠn, wereferto aformula
h(z1, . . . , zn) =a+
(A−z1Y1− · · · −znYn)−1v, v
H
where aisaconstant,HisaHilbertspace,A isadenselydefinedself-adjointoperator onH,Y1,. . . ,Yn arepositivecontractionsonHsummingto1 andv isavectorinH,as
aNevanlinnarepresentation of type2.
Theorem 1.7. A function hdefined on Πn has aNevanlinna representation of type 2if and onlyif h∈ Ln and
lim inf
y→∞ yImh(iy, . . . , iy)<∞. (1.11)
A Nevanlinnarepresentation of type3 ofafunctionhdefinedonΠn isoftheform
h(z) =a+(1−iA)(A−zY)−1(1 +zYA)(1−iA)−1v, v
for allz∈Πn
for some real a, some self-adjoint operatorA and some vector v, where Y1,. . . ,Yn are
operators asinequation(1.4)aboveand zY =z1Y1+· · ·+znYn.
Theorem 1.8. A function hdefined on Πn has aNevanlinna representation of type 3if and onlyif h∈ Ln and
lim inf
y→∞
1
y Imh(iy, . . . , iy) = 0.
Finally,Nevanlinna representationsoftype 4 aregivenbytheformula
h(z) =a+M(z)v, v, (1.12)
where a∈RandM(z) isanoperatoroftheform
−i 0 0 1−iA
1 0 0 A
−zP
0 0 0 1
−1 zP
1 0 0 A
+
0 0 0 1
−i 0 0 1−iA
−1 ,
(1.13)
acting onanorthogonaldirectsumofHilbertspacesN ⊕ M.Inequation(1.12),v isa vectorinN ⊕ M.Inequation(1.13),Aisadensely-definedself-adjointoperatoracting onMand zP is theoperatoractingonN ⊕ Mviatheformula
zP =
ziPi
where P1,. . . ,Pn arepairwise orthogonalprojectionsactingonN ⊕ Mthatsumto1.
Theorem1.9.LethbeafunctiondefinedonΠn.ThenhhasaNevanlinnarepresentation
A weaker, “generic” version of Theorem 1.9 appeared in [4,Theorem 6.9], where it wasusedtoshowthatelementsinLn arelocally operator-monotone.
It turns out that for 1 ≤ k ≤ 4, if h is a function on Πn and hhas aNevanlinna
representationof typek, thenfork≤j ≤4,halso hasaNevanlinna representation of typej.Thus,itisnaturaltodefinethetypeofafunctioninLn tobethesmallestksuch
thathhasaNevanlinna representationoftypek.
Forh∈ Ln thetypeofhcanbecharacterizedinfunction-theoretictermsthroughthe
useofageometricideadue toCarathéodory. Acarapoint forafunctionϕintheSchur classSn isapointτ∈Tsuchthat
lim inf
λ→τ
1− |ϕ(λ)| 1− λ∞ <∞,
where
λ∞= max
1≤i≤n|λi|.
Carathéodory introduced this notion in one variable in [9], along the way to refining earlierresultsofJulia[14].Thefollowing wasCarathéodory’smain result;thenotation
λ→nt τ meansthatλtendsnontangentially toτ.
Theorem 1.10. Letϕ∈ S1, τ ∈ T.If τ is acarapoint for ϕ, then ϕ isnontangentially
differentiableatτ,thatis, thereexistvaluesϕ(τ)andϕ(τ) suchthat
lim
λ→ntτ
ϕ(λ)−ϕ(τ)−ϕ(τ)(λ−τ)
λ−τ = 0.
In particular, if τ is a carapoint for ϕ then there exists a unique point ϕ(τ)∈ T such that ϕ(λ)→ϕ(τ)asλ→nt τ.
Inseveral variables, carapointshavebeen studiedin[1,13,3]. Thestrong conclusion of nontangential differentiability islost inseveral variables; however, at acarapoint τ, therestillexists aunimodularnontangentiallimitϕ(τ).
As the point χ = (1,. . . ,1) is transformed to the point ∞ = (∞,. . . ,∞) by equa-tion(1.6),itisnaturaltosaythatafunctionh∈ Lnhasacarapointat∞iftheassociated
Schurfunction ϕ, given by thetransformation in equation(1.7), has acarapointat χ, andinthatcasetodefineh(∞) by
h(∞) =i1 +ϕ(χ)
1−ϕ(χ). (1.14)
Theorem 1.11.Forafunctionh∈ Ln,
(1) hisof type1if andonlyif ∞isa carapointofhandh(∞)= 0;
(2) hisof type2if andonlyif ∞isa carapointofhandh(∞)∈R\ {0};
(3) hisof type3if andonlyif ∞isnot acarapointof h;
(4) hisof type4if andonlyif ∞isa carapointofhandh(∞)=∞.
The paper is structuredas follows. As is clearfrom theformulae used to define the various Nevanlinna representations, Nevanlinna representations are generalizations of theresolventofaself-adjointoperator.Thesestructuredresolvents,studiedinSections2
and3,areanalyticoperator-valuedfunctionsonthepolyhalf-plane Πnwithnon-negative
imaginary part,fullyanalogoustothefamiliar resolventoperator.There arealso struc-tured resolvent identitiesforthem,studiedinSection10ofthepaper.
InmoderntextsNevanlinna’srepresentationisderivedfromtheHerglotz Representa-tionwiththeaidoftheCayleytransform[16,10].InSection4weintroducethen-variable
strong Herglotz classandthen proveTheorem 1.9by applyingtheCayleytransformto Theorem 1.8of[2].
In Section5wederive theNevanlinna representations oftypes 3, 2,and1, weshow howtheyarisenaturallyfromtheunderlyingHilbertspacegeometryandweproveslight strengthenings of Theorems 1.6, 1.7 and 1.8. In Section 6 we give function-theoretic conditionsforafunctionh∈ Ln topossessarepresentationof agiventype.
InSection7weintroducethenotionofcarapointsforfunctionsinthePickclassand in Section8we establish thecriteria inTheorem 1.11 forthe type ofa functionusing thelanguageofcarapoints.
In Section 9 we give the growth estimates for functions in Ln that flow from our
analysis of structured resolvents, and inSection 10 we present resolvent identities for structuredresolvents.
Results related to ours from a system-theoretic perspective have been obtained in recentworks ofJ.A. BallandD. Kalyuzhnyi-Verbovetzkyi[6,7].Seealso[8],whereKrein space methodsareappliedtosimilarproblems.
2. Structuredresolventsof operators
The resolvent operator(A−z)−1 of a densely defined self-adjoint operator A on a
Hilbertspaceplays aprominentrole inspectraltheory. Ithasthefollowing properties.
(1) Itisananalyticbounded operator-valuedfunctionofz intheupperhalf-plane Π; (2) itsatisfiesthegrowth estimate(A−z)−1≤1/Imzforz∈Π;
(3) (A−z)−1 hasnon-negativeimaginary partforallz∈Π;
(4) itsatisfiesthe“resolventidentity”.
the polyhalf-plane Πn. Because of the additional complexities in several variables we encounterthree different typesofresolvent; allof them havethefour listedproperties, withveryslightmodifications, andthereforedeservethenamestructuredresolvent.
For any Hilbert space H, a positive decomposition of H will mean an n-tuple Y = (Y1,. . . ,Yn) of positive contractions on H that sum to the identity operator. For any
z= (z1,. . . ,zn)∈Cn andanyn-tupleT = (T1,. . . ,Tn) ofboundedoperatorswedenote
by zT the operator
jzjTj. Here each Tj is a bounded operator from H1 to H2, for
someHilbertspacesH1,H2,so thatzT isalsoabounded operatorfrom H1 toH2.
Definition 2.1. Let A be a closed densely defined self-adjoint operator on a Hilbert space H and let Y be a positive decomposition of H. The structured resolvent of A
oftype 2 correspondingtoY istheoperator-valuedfunction
z→(A−zY)−1: Πn → L(H).
Thefollowingobservationisessentially[4,Lemma6.25].
Proposition 2.2. ForA and Y asin Definition 2.1 the structuredresolvent (A−zY)−1
iswelldefined onΠn andsatisfies,forallz∈Πn,
(A−zY)−1 ≤
1 minjImzj
. (2.1)
Moreover
Im(A−zY)−1
= (A−z∗Y)−1(ImzY) (A−zY)−1
= (A−zY)−1(ImzY) (A−z∗Y)−1
≥0. (2.2)
Therangeof thebounded operator(A−zY)−1 isofcourseD(A),thedomainofA.
ProofofProposition2.2. ForanyvectorξinthedomainofA,
(A−zY)ξ ξ ≥ | (A−zY)ξ, ξ |
≥ |Im(A−zY)ξ, ξ |
=(ImzY)ξ, ξ
=
j
(Imzj)Yjξ, ξ
≥(min
j Imzj)
j
Yjξ, ξ
= (min
j Imzj)ξ
ThusA−zY haslowerboundminjImzj >0,andsohasaboundedleftinverse.A similar
argument with z replaced by ¯z shows that (A−zY)∗ also hasa bounded left inverse,
and soA−zY hasabounded inverseandtheinequality(2.1)holds.
Theidentities(2.2)areeasy. 2
Resolvents of type 2 are thesimplest several-variableanalogues of the familiar one-variable resolventbuttheyarenotsufficientfortheanalysisoftheseveral-variablePick class.Tothisendweintroducetwofurthergeneralizations.Letusfirstrecallsomebasic factsaboutclosedunboundedoperators.
Lemma2.3.LetT beacloseddenselydefinedoperatoronaHilbertspaceH,withdomain D(T).The operator1+T∗T is abijectionfromD(T∗T)toH,andtheoperators
B def= (1 +T∗T)−1, Cdef= T(1 +T∗T)−1
are everywhere defined and contractive on H. Moreover B is self-adjoint and positive, and ranC⊂ D(T∗).
Proof. All these statements are proved in [21, Sections 118, 119], although the final statement aboutranC is notexplicitlystated. We must showthatfor allv ∈ H there exists y∈ Hsuchthat,forallh∈ H,
T h, Cv=h, y.
It is straightforwardto check thatthis relation holds for y =v−Bv, and so ranC ⊂
D(T∗). 2
Definition2.4.LetAbeacloseddenselydefinedself-adjointoperatoronaHilbertspace
H andlet Y be apositivedecompositionof H. Thestructured resolvent of A of type 3 corresponding toY istheoperator-valuedfunctionM: Πn → L(H) givenby
M(z) = (1−iA)(A−zY)−1(1 +zYA)(1−iA)−1. (2.3)
Wedenotethe1 normonCn by· 1.NotethatzY≤ z1forallz∈Cn and all
positivedecompositionsY.
Proposition2.5.ForAandY asinDefinition2.4thestructuredresolventM(z)oftype3
given by equation (2.3) is welldefined asa bounded operator on H forall z ∈Πn and
satisfies
M(z) ≤(1 + 2z1)
1 + 1 +z1 minjImzj
Proof. Since
1 +zYA= 1−izY +izY(1−iA) :D(A)→ H
and(1−iA)−1isacontractiononallofH,withrangeD(A),theoperator(1+zYA)(1−
iA)−1 iswell definedas anoperatoronHand
(1 +zYA)(1−iA)−1=(1−izY)(1−iA)−1+izY
≤ 1−izY+zY
≤1 + 2zY
≤1 + 2z1. (2.5)
Similarly(1−iA)(A−zY)−1iswell definedonH,andsince
i(A−zY) =−(1−iA) + (1−izY) :D(A)→ H
wehave
i=−(1−iA)(A−zY)−1+ (1−izY)(A−zY)−1:H → H.
Thus,byvirtueofthebound(2.1),
(1−iA)(A−zY)−1=i−(1−izY)(A−zY)−1
≤1 +1−izY (A−zY)−1
≤1 + 1 +z1 minjImzj
. (2.6)
Oncombiningtheestimates(2.6)and(2.5)weobtainthebound(2.4). 2
Thefollowingalternativeformula forthestructuredresolventoftype 3, validonthe densesubspaceD(A) ofH,allowsusto showthatImM(z)≥0.
Proposition2.6. ForA andY asin Definition 2.4andz∈Πn
M(z)|D(A) = (1−iA)(A−zY)−1−A(1 +A2)−1
(1 +iA) (2.7)
= (1−iA)(A−zY)−1(1 +iA)−A:D(A)→ H. (2.8)
Moreover,forevery v∈ D(A),
ImM(z)v, v=(1−iA)(A−zY∗)−1(ImzY)(A−zY)−1(1 +iA)v, v
Proof. By Lemma 2.3 the operator A(1+A2)−1 is contractive on H and has range containedinD(A).OnD(A2) wehavetheidentity
1 +zYA= 1 +A2−(A−zY)A.
Since (1+A2)−1mapsHinto D(A2) wehave
(1 +zYA)(1 +A2)−1= 1−(A−zY)A(1 +A2)−1:H → H,
and therefore
(A−zY)−1(1 +zYA)(1 +A2)−1= (A−zY)−1−A(1 +A2)−1:H → D(A).
(2.10)
Clearly
(1 +A2)−1(1 +iA) = (1−iA)−1 onD(A)
andso,onmultiplyingequation(2.10)fore-and-aft by1±iA,wededucethat,asoperators from D(A) toH,
M(z)|D(A) = (1−iA)(A−zY)−1(1 +zYA)(1−iA)−1
= (1−iA)(A−zY)−1(1 +zYA)(1 +A2)−1(1 +iA)
= (1−iA)(A−zY)−1−A(1 +A2)−1
(1 +iA).
This establishesequation(2.7).
Theexpression (2.8)follows fromequation(2.7)since
(1−iA)A(1 +A2)−1(1 +iA) =A onD(A).
Byequation(2.8)wehave,foranyz∈Πn andv∈ D(A),
ImM(z)v, v= Im(1−iA)(A−zY)−1(1 +iA)v, v
−ImAv, v
= Im(A−zY)−1(1 +iA)v,(1 +iA)v
and hence,byequation (2.2),
ImM(z)v, v=(A−zY∗)−1(ImzY)(A−zY)−1(1 +iA)v,(1 +iA)v
,
and soequation(2.9)holds. 2
For, by Propositions 2.5 and 2.6, M(z) is a bounded operator on H, and ImM(z)v,v ≥ 0 for v ∈ D(A). The conclusion follows by the density of D(A) and continuity.
InthecaseofboundedA thereisyetanotherexpressionforthestructuredresolvent oftype 3.
Proposition 2.8. If A is a bounded self-adjoint operator on H and Y is a positive de-compositionof Hthen,forz∈Πn,
M(z) = (1 +iA)−1(1 +AzY)(A−zY)−1(1 +iA). (2.11)
Proof. SinceA isboundeditisdefinedonallofH.Wehave
1 +AzY = 1 +A2−A(A−zY)
andhence
(1 +AzY)(A−zY)−1= (1 +A2)(A−zY)−1−A.
Thus
(1 +iA)−1(1 +AzY)(A−zY)−1(1 +iA) = (1−iA)(A−zY)−1(1 +iA)−A
=M(z)
byequation(2.8). 2
Remark2.9.InthecaseofunboundedAtheexpression(2.11)forM(z) isvalidwherever itisdefined,butitisnottobeexpectedthatthiswillbeadensesubspaceofHingeneral.
Here are two examplesof structured resolventsof type 3, oneon C2 and oneon an
infinite-dimensionalspace.
Example2.10.Let
H=C2, A=
1 0 0 −1
, Y1= 12
1 1 1 1
, Y2= 1−Y1, Y = (Y1, Y2).
Then
M(z) = (1−iA)(A−zY)−1(1 +zYA)(1−iA)−1
= 1
1−z1z2
(1 +z1)(1 +z2) −i(z1−z2) i(z1−z2) −(1−z1)(1−z2)
Example 2.11.LetH=L2(R),letA betheoperationofmultiplication bythe indepen-dent variablet and letY = (P,Q) whereP,Q aretheorthogonal projectionoperators ontothesubspacesofevenandoddfunctionsrespectivelyinL2.Thus
P f(t) = 12{f(t) +f(−t)}, Qf(t) =12{f(t)−f(−t)}.
LetY = (Q,P).Notethat
P A=AQ, QA=AP
and hence
zYA=AzY, zYA=AzY, zYzY =z1z2=zYzY.
It followsthatzY andzY commutewithA2,and itmaybecheckedthat
(A−zY)−1= (A2−z1z2)−1(zY+A) = (zY+A)(A2−z1z2)−1
and hence
(A−zY)−1(1 +zYA) = (A2−z1z2)−1
(1 +A2)zY+ (1 +z1z2)A
.
AstraightforwardcalculationnowshowsthatthestructuredresolventM(z) ofA corre-spondingto Y isgivenby
(M(z)f)(t) =
1
2(z1+z2)(1 +t
2) + (1 +z 1z2)t
f(t) +12(z2−z1)(1−it)2f(−t) t2−z
1z2
forallz∈Π2,f ∈L2(R) andt∈R.Inparticular,wenote forfutureusethatiff is an
even function,
(M(z)f)(t) =t(1 +z1z2) + (1−it)(itz1+z2)
t2−z 1z2
f(t). (2.12)
3. Thematricialresolvent
Thethirdandlastformofstructuredresolventthatweconsiderhasa2×2 matricial form. Aswill becomeclear,this extracomplication isneededfor thedescriptionof the mostgeneraltypeoffunctionintheseveral-variableLoewnerclass.
Byanorthogonal decomposition ofaHilbertspace Hweshallmean ann-tupleP = (P1,. . . ,Pn) of orthogonal projection operators with pairwise orthogonal ranges such
Proposition 3.1. LetHbe theorthogonal directsum ofHilbert spaces N,M,letA be a denselydefinedself-adjointoperatoronMwithdomainD(A)andletP beanorthogonal decomposition of H. For every z ∈ Πn the operator on H given with respect to the decompositionN ⊕ Mby thematricialformula
M(z) =
−i 0 0 1−iA
1 0 0 A
−zP
0 0 0 1
−1
×
zP
1 0 0 A
+
0 0 0 1
−i 0 0 1−iA
−1
(3.1)
isaboundedoperatordefined onallof H,and
M(z) ≤(1 +√10z1)
1 +1 +
√
2z1
minjImzj
. (3.2)
Proof. Letz∈Πn. LettheprojectionP
j haveoperatormatrix
Pj=
Xj Bj
Bj∗ Yj
(3.3)
withrespectto thedecompositionH=N ⊕ M.Then
X = (X1, . . . , Xn), Y = (Y1, . . . , Yn)
arepositivedecompositionsofN,Mrespectively,and
B = (B1, . . . , Bn), B∗= (B1∗, . . . , Bn∗)
aren-tuplesofcontractionssummingto0,fromMtoN andfromN toMrespectively. SincetheBj arecontractionswehave
zB ≤ z1.
Foranyz∈Cn,
zP =
zX zB
zB∗ zY
. (3.4)
Considerthethirdandfourthfactorsintheproductontherighthandsideofequation
(3.1);theproductofthesetwofactorsiswelldefinedasanoperatoronHsince(1−iA)−1
mapsMtoD(A).It isevenabounded operator,since,byvirtueofequation(3.4),
zP
1 0 0 A
+
0 0 0 1
−i 0 0 1−iA
−1
=
izX zBA(1−iA)−1
izB∗ (1 +zYA)(1−iA)−1
.
Since
A(1−iA)−1=i1−(1−iA)−1 ≤2
we canimmediately see thattheoperator(3.5) isbounded. We cangetan estimateof thenorm oftheoperatormatrix(3.5)ifwereplaceeachofthefour operatorentries by anupperbound foritsnorm. Wefindthat
zP 1 0 0 A + 0 0 0 1
−i 0 0 1−iA
−1
≤
z1 2z1
z1 1 + 2z1
≤1 +z1
1 2
1 2
= 1 +√10z1. (3.6)
Now considerthesecondfactorinthedefinition(3.1)ofM(z).Wefindthat
1 0 0 A
−zP
0 0 0 1 −1 =
1 −zB
0 A−zY
−1
=
1 zB(A−zY)−1
0 (A−zY)−1
, (3.7)
whichmapsHintoN ⊕ D(A).Hencetheproductofthefirsttwofactorsintheproduct ontherighthandsideofequation(3.1)is
−i 0 0 1−iA
1 0 0 A
−zP
0 0 0 1 −1 =
−i −izB(A−zY)−1
0 (1−iA)(A−zY)−1
. (3.8)
Since
(1−iA)(A−zY)−1=(1−izY)(A−zY)−1−i
≤1 +1−izY (A−zY)−1
≤1 + 1 +z1 minjImzj
we deducefromequation(3.8)that
−i 0 0 1−iA
1 0 0 A
−zP
0 0 0 1 −1 ≤
1 z1(A−zY)−1
0 1 + (1 +z1)(A−zY)−1
≤1 +
0 z1
0 1 +z1
0 0
0 (A−zY)−1
≤1 + 1 +
√
2z1
minjImzj
. (3.9)
Oncombiningtheestimates(3.9)and(3.6)weobtainthebound(3.2)forM(z). 2
Remark 3.2. On multiplying together the expressions (3.8) and (3.5) we obtain the formula
M(z) =
zX+zB(A−zY)−1zB∗ −izB(A−zY)−1(1 +iA)
i(1−iA)(A−zY)−1zB∗ (1−iA)(A−zY)−1(1 +zYA)(1−iA)−1
.
Noticeinparticularthatthe(2,2) entry(thatis,thecompressionofM(z) toM)isthe structuredresolventofA oftype 3 correspondingtoY, thecompressionof P to M,as inequation (2.3).
Definition3.3. LetH bethe orthogonaldirectsumof HilbertspacesN,M, letAbe a denselydefinedself-adjointoperatoronMwithdomainD(A) andletPbeanorthogonal decompositionof H. Thestructured resolvent of A of type 4 correspondingto P is the operator-valuedfunctionM : Πn→ L(H) givenbyequation (3.1).
We shall also refer to M(z) as the matricial resolvent of A with respect to P. The important property thatImM(z)≥ 0 is notat once apparent from the formula (3.1); as with structuredresolvents of type 3, there are alternativeformulae from which this property is more easily shown. Once again the alternatives suffer theminor drawback thattheygiveM(z) onlyonadensesubspaceofH.
Proposition3.4. With thenotation of Definition 3.3,asoperatorson N ⊕ D(A),
M(z) =
−i 0 0 1−iA
1 0
0 A(1 +A2)−1
zP+
0 0
0 (1 +A2)−1 ×
1 0 0 A − 0 0 0 1 zP −1 i 0
0 1 +iA
(3.10)
=
−i 0 0 1−iA
1 0 0 0
zP+
0 0 0 1 1 0 0 A − 0 0 0 1 zP −1 i 0
0 1 +iA − 0 0 0 A (3.11) =
−i 0 0 1−iA
1 0 0 A
−zP
0 0 0 1 −1 zP 1 0 0 0 + 0 0 0 1 i 0
0 1 +iA − 0 0 0 A (3.12)
M(z)−M(w)∗=
−i 0 0 1−iA
1 0 0 A
−w∗P
0 0 0 1
−1
×
(zP−w∗P)
1 0 0 A − 0 0 0 1 zP −1 i 0
0 1 +iA
(3.13)
on N ⊕ D(A).
Proof. ByLemma 2.3theoperators(1+A2)−1 and
Cdef= Im(1−iA)−1=A(1 +A2)−1
are self-adjointcontractionsdefinedonallofM.Furthermore,
ran(1 +A2)−1=D(A2), ranC⊂ D(A).
Weclaimthat,asoperators onN ⊕ D(A),
1 0 0 A
−zP
0 0 0 1 −1 zP 1 0 0 A + 0 0 0 1 =
1 0 0 C
zP +
0 0
0 (1 +A2)−1
1 0 0 C − 0 0
0 (1 +A2)−1
zP −1 . (3.14) Wehave zP 1 0 0 A + 0 0 0 1 1 0 0 C − 0 0
0 (1 +A2)−1
zP = 0 0 0 C
+zP
1 0 0 AC − 0 0
0 (1 +A2)−1
zP−zP
0 0 0 C zP = 0 0 0 C
+zP
1 0 0 AC
−1 +
1−
0 0
0 (1 +A2)−1 zP−zP
0 0 0 C zP = 0 0 0 C
−zP
0 0
0 (1 +A2)−1
+ 1 0 0 AC
zP−zP
0 0 0 C zP = 1 0 0 A
−zP
0 0 0 1 1 0 0 C
zP +
0 0
0 (1 +A2)−1 .
This is an identity between operators on H, in both cases a composition H → N ⊕
Oncombiningequations(3.1)and(3.14)wededucethat
M(z) =
−i 0 0 1−iA
1 0 0 C
zP +
0 0
0 (1 +A2)−1 ×
1 0 0 C − 0 0
0 (1 +A2)−1
zP
−1
−i 0 0 1−iA
−1 .
Since
−i 0 0 1−iA
−1
=
1 0
0 1 +A2
−1
i 0
0 1 +iA
and
1 0
0 1 +A2
1 0 0 C − 0 0
0 (1 +A2)−1
zP =
1 0 0 A − 0 0 0 1 zP,
wededucefurtherthat
M(z) =
−i 0 0 1−iA
1 0 0 C
zP +
0 0
0 (1 +A2)−1 ×
1 0 0 A − 0 0 0 1 zP −1 i 0
0 1 +iA
, (3.15)
whichprovesequation(3.10).Itisstraightforwardtoverifythat
1 0 0 C
zP+
0 0
0 (1 +A2)−1
1 0 0 A − 0 0 0 1 zP −1 (3.16) = 1 0 0 0 zP+
0 0 0 1 1 0 0 A − 0 0 0 1 zP −1 − 0 0
0 A(1 +A2)−1
.
(3.17)
Clearly
−i 0 0 1−iA
0 0
0 A(1 +A2)−1
i 0
0 1 +iA = 0 0 0 A ,
andsoonsuitablypre- andpost-multiplyingequation(3.16),weobtainequation(3.11). Toproveequation (3.12),checkfirstthat
1 0 0 A
−zP
0 0 0 1 1 0 0 0 zP +
0 0 0 1 =
as operatorsonN ⊕ D(A).Itfollows that
1 0 0 0
zP+
0 0 0 1 1 0 0 A − 0 0 0 1 zP −1 = 1 0 0 A
−zP
0 0 0 1 −1 zP 1 0 0 0 + 0 0 0 1
as operators from HtoN ⊕ D(A). Oncombiningthis relationwithequation (3.11)we derive theexpression(3.12)forM(z)|N ⊕ D(A).
Wenow derivetheidentity(3.13).Let
D=
i 0
0 1 +iA
and considerz,w∈Πn.Byequation(3.10)
M(z) =D∗W(z)D (3.18)
onN ⊕ D(A),where
W(z) =R(z)S(z)−1−
0 0
0 A(1 +A2)−1
(3.19)
and
R(z) =
1 0 0 0
zP+
0 0 0 1
, S(z) =
1 0 0 A − 0 0 0 1 zP.
WehaveseenthatS(z) isinvertibleforanyz∈Πn,sothatW(z) isaboundedoperator onH.Clearly
M(z)−M(w)∗=D∗R(z)S(z)−1−S(w)∗−1R(w)∗D
=D∗S(w)∗−1(S(w)∗R(z)−R(w)∗S(z))S(z)−1D.
Here
S(w)∗R(z)−R(w)∗S(z) =
1 0 0 0
zP+
0 0 0 A
−w∗P 0 0 0 1 −
wP∗ 1 0 0 0 + 0 0 0 A − 0 0 0 1 zP
=zP −wP∗.
M(z)−M(w)∗=D∗S(w)∗−1(zP−w∗P)S(z)−1D,
whichisequation(3.13). 2
The nextresult shows thatthe matricial resolvent belongs not justto theoperator Pickclass, butto thesmalleroperatorLoewner class.
Proposition 3.5. With thenotation of Definition 3.3,there exists an analytic operator-valuedfunction F: Πn→ L(H) suchthatforallz,w∈Πn,
M(z)−M(w)∗=F(w)∗(z−w¯)PF(z) (3.20)
onH.
Proof. The identity (3.13) shows that such a relation holds on N ⊕ D(A); we must extendittoallofH.WritePj asanoperatormatrixwithrespecttothedecomposition
H=N ⊕M,asinequation(3.3).ThenzP hasthematricialexpression(3.4).Forz∈Πn
let
F(z) =
1 0 0 A
−
0 0 0 1
zP
−1
i 0
0 1 +iA
.
ThenF(z) isanoperatorfrom N ⊕ D(A) to H, andwefindthat
F(z) =
1 0
−zB∗ A−zY
−1
i 0
0 1 +iA
=
i 0
i(A−zY)−1zB∗ (A−zY)−1(1 +iA)
:N ⊕ D(A)→ H.
Let
F(z) =
i 0
i(A−zY)−1zB∗ i+ (A−zY)−1(1 +izY)
:N ⊕ M → H. (3.21)
Since
(A−zY)−1(1 +iA) =i+ (A−zY)−1(1 +izY)
onN ⊕ D(A) andtheright handside ofthelast equationisaboundedoperatoronall ofH,itisclearthat,foreveryz∈Πn,F(z) isacontinuousextensiontoHofF(z) and
isabounded operator.FurthermoreF isanalyticonΠn.
ByProposition 3.4, equation (3.13), therelation(3.20) holds onthe densesubspace
N ⊕ D(A) ofHforeveryz,w∈Πn.Sincetheoperatorsonbothsidesofequation(3.20)
Corollary 3.6. A matricial resolvent has a non-negative imaginary part at every point of Πn.
Proof. In the notation of Proposition 3.5, on choosing w = z in equation (3.20) and dividingby2iweobtaintherelation
ImM(z) =F(z)∗(ImzP)F(z)
onH.Wehave
ImzP =
j
(Imzj)Pj≥0,
and soImM(z)≥0 onHforallz∈Πn. 2
Here isaconcrete exampleof amatricialresolvent.
Example 3.7.Thefunction
M(z) = 1
z1+z2
2z1z2 i(z1−z2)
−i(z1−z2) −2
(3.22)
is thematricial resolventcorrespondingto
H=C2, N =M=C, A= 0 onC, P1= 12
1 1 1 1
, P2= 1−P1.
4. Nevanlinnarepresentationsoftype 4
InthissectionwederiveamultivariableanalogofthemostgeneralformofNevanlinna representationforfunctionsintheone-variablePickclass(Theorem 1.2).Westartwitha multivariableHerglotztheorem[2,Theorem1.8].Weshallsay(followingG. Herglotz[12]) thatananalyticoperator-valuedfunctionF onDn isaHerglotzfunctionifReF(λ)≥0 forallλ∈Dn. Forpresent purposesweneedthefollowingmodification ofthenotion.
Definition 4.1. An analytic functionF : Dn → L(K), where K is a Hilbert space, is a
strong Herglotz function if, forevery commutingn-tuple T = (T1,. . . ,Tn) ofoperators
onaHilbertspaceandfor0≤r <1,ReF(rT)≥0.
In[2]thesefunctionswerecalledFn-Herglotz functions.TheclassofstrongHerglotz
Theorem 4.2. Let K be a Hilbert space and let F : D2 → L(K) be a strong Herglotz functionsuchthatF(0)= 1.ThereexistaHilbertspaceH,anorthogonaldecomposition
P of H, an isometric linear operatorV : K → H anda unitary operator U on H such that,forallλ∈Dn,
F(λ) =V∗1 +U λP
1−U λP
V. (4.1)
Conversely, every function F : Dn → L(K) expressible in the form (4.1) for some
H, P, V and U with the stated properties is a strong Herglotz function and satisfies
F(0)= 1.
NotethatλP =
jλjPj hasoperatornormatmostλ∞<1 forλ∈Dn,andhence
equation(4.1)doesdefineF asananalyticoperator-valuedfunctiononDn.
Onspecializing toscalar-valued functionsinthen-variable Herglotz class weobtain thefollowingconsequence.
Corollary 4.3. Letf be a scalar-valued strong Herglotz function on Dn. There exists a HilbertspaceH,a unitaryoperatorLonH,anorthogonaldecompositionP of H,a real numbera andavectorv∈ Hsuchthat, forallλ∈Dn,
f(λ) =−ia+(L−λP)−1(L+λP)v, v
. (4.2)
Conversely,foranyH,L,P,aandv withthepropertiesdescribed,equation(4.2)defines
f asan n-variablestrongHerglotz function.
Again,therighthandsideof equation(4.2)isananalytic functionofλ∈Dn since
(L−λP)−1=L−1(1−λPL−1)−1
isabounded operatorandisanalytic inλ.
Definition4.4. ANevanlinna representationof type 4 ofafunctionh: Πn→Cconsists ofanorthogonallydecomposedHilbertspaceH=N ⊕ M,a self-adjointdenselydefined operatorA onM, anorthogonaldecomposition P ofH, a real numberaand avector
v∈ Hsuchthat
h(z) =a+M(z)v, v (4.3)
forallz∈Πn,whereM(z) isthestructuredresolventofAoftype 4 correspondingtoP
(givenbytheformula(3.1)).
results inrepresentation theorems fora subclass of thePick class Pn. Recall from the
introduction:
Definition 4.5. The Loewner class Ln is the set of analytic functions h on Πn with
theproperty thatthere existnpositivesemi-definitefunctionsA1,. . . ,An onΠn×Πn,
analytic inthefirstargument,suchthat
h(z)−h(w) =
n
j=1
(zj−wj)Aj(z, w)
forallz,w∈Πn.
AfunctionhonΠnbelongstoL
nifandonlyifitcorrespondsunderconjugationbythe
CayleytransformtoafunctionintheSchur–Aglerclassofthepolydisc[4,Lemma 2.13]. Another characterization: h ∈ Ln if and only if, for every commuting n-tuple T of
bounded operators with strictly positiveimaginary parts,h(T) has positiveimaginary part.
We can now prove Theorem 1.9 from the introduction: a function h defined on Πn
has aNevanlinna representation oftype 4if andonly ifh∈ Ln.
Proof ofTheorem 1.9. Let h∈ Ln. Define ann-variableHerglotz function f :Dn →C
by
f(λ) =−ih(z) (4.4)
where
zj =i
1 +λj
1−λj
forj= 1, . . . , n. (4.5)
Whenλ∈DnthepointzbelongstoΠn,andsof(λ) iswelldefined,andsinceImh(z)≥0
we haveRef(λ)≥0,so thatf isindeedaHerglotz function.Infactf iseven astrong Herglotz function: since h ∈ Ln, the function ϕ ∈ Sn corresponding to h lies in the
Schur–Agler class of the polydisc, and so f = (1+ϕ)/(1−ϕ) is a strong Herglotz function.
By Corollary 4.3 there exist a real number a, a Hilbert space H, a vector v ∈ H, a unitary operatorLonHand anorthogonal decompositionP onHsuchthat,for all
z∈Πn,
h(z) =if(λ) =a+i(L−λ)−1(L+λ)v, v
=a+i[L−(z−i)(z+i)−1]−1[L+ (z−i)(z+i)−1]v, v. (4.6)
Here and intherestof thissection z, λareidentified withtheoperators zP,λP onH,
λ= z−i
z+i
ismeaningfulandvalid. Forz∈Πn let
M(z) =i(L−λ)−1(L+λ) =i
L−z−i z+i
−1
L+z−i
z+i . (4.7)
Since L is unitary on H and λ ∈ Dn, the operator M(z) is bounded on H for every
z∈Πn and, byequation (4.6), wehave
h(z) =a+M(z)v, v (4.8)
for all z ∈ Π2. Theorem 1.9 will follow provided we can show that M(z) is given by
equation(3.1)forasuitableself-adjoint operatorA. Observethat
M(z) =i((z+i)L−(z−i))−1((z+i)L+ (z−i))
=i(z(L−1) +i(L+ 1))−1(z(L+ 1) +i(L−1)). (4.9)
We wish to take outa factor 1−L from both factors inequation (4.9), but this may beimpossiblesince1−Lcanhaveanonzerokernel.Accordinglywedecompose Hinto
N ⊕ Mwhere N = ker(1−L),M=N⊥.With respect tothis decomposition wecan writeLas anoperatormatrix
L=
1 0 0 L0
,
whereL0 isunitaryandker(1−L0)={0}. Substitutinginto equation(4.9)wehave
M(z) =i
z
0 0
0 L0−1
+i
2 0
0 L0+ 1
−1 z
2 0
0 L0+ 1
+i
0 0
0 L0−1
z = −z 0 0
0 1−L0
+
2i 0 0 i(1 +L0)
−1 z
2i 0 0 i(1 +L0)
+
0 0
0 1−L0
(4.10)
Formallywemaynowwrite
M(z) =
−1
2i 0
0 (1−L0)−1 −z
0 0 0 1 + 1 0
0 i1+L0 1−L0
−1 × z 1 0
0 i1+L0 1−L0 + 0 0 0 1
2i 0 0 1−L0
butwhereasequation(4.10)isarelationbetweenboundedoperatorsdefinedonallofH, equation(4.11)involvesunbounded,partiallydefinedoperatorsandwemustverifythat theproduct ofoperatorsontherighthandsideismeaningful.
Let
A=i1 +L0
1−L0 .
Since L0 is unitary on M and ker(1−L0) = {0}, the operator A is self-adjoint and
densely defined onM [21, Section 121]. The domain D(A) ofA is the dense subspace ran(1−L0) ofM.Itfollows fromthedefinitionofAthat
(1−L0)−1=12(1−iA), (4.12)
whichisanequation betweenbijectiveoperators fromD(A) toM.Likewise
1 +L0=−2iA(1−iA)−1:M → D(A) (4.13)
are boundedoperators.
Letuscontinuethecalculationfromthefirstfactorontherighthandsideofequation
(4.10).Sinceker(1−L0)={0},therighthandsideoftherelation
−z
0 0
0 1−L0
+
2i 0 0 i(1 +L0)
= −z 0 0 0 1 + 1 0 0 A
2i 0 0 1−L0
comprisesabijectivemapfrom HtoN ⊕ D(A) followedbyabijectionfrom N ⊕ D(A) toH(recalltheequation(3.7)).Wemaythereforetakeinversesintheequationtoobtain
−z
0 0
0 1−L0
+
2i 0 0 i(1 +L0)
−1
=
−1
2i 0
0 (1−L0)−1
1 0 0 A −z 0 0 0 1 −1 = −1
2i 0
0 12(1−iA)
1 0 0 A −z 0 0 0 1 −1 (4.14)
as operatorsonN ⊕ D(A).
Similarreasoningappliestotheequation
z
2i 0 0 i(1 +L0)
+
0 0
0 1−L0
= z 1 0 0 A + 0 0 0 1
2i 0 0 1−L0
= z 1 0 0 A + 0 0 0 1 −1
2i 0
0 1
2(1−iA)
−1
it is valid as an equation between operators on H. The right hand side comprises an operatorfrom H to N ⊕ D(A) followedby an operator from N ⊕ D(A) to H, and so bothsidesoftheequationdenote anoperatoronH.
Oncombiningequations(4.10),(4.14) and(4.15)weobtain
M(z) =
−1
2i 0
0 12(1−iA)
1 0 0 A
−z
0 0 0 1
−1
×
z
1 0 0 A
+
0 0 0 1
−1
2i 0
0 1
2(1−iA)
−1 .
Pre-multiply this equationby 2 andpost-multiply by 12 todeduce thatM(z) isindeed thestructuredresolventofAoftype 4 correspondingtoP,asdefinedinequation(3.1). Thustheformula(4.8)isaNevanlinnarepresentation ofhoftype 4.
Conversely,let h∈ Ln haveatype 4 representation (4.3). ByProposition 3.5there
existsananalyticoperator-valuedfunctionF : Πn→ L(H) suchthat,forallz,w∈Πn,
M(z)−M(w)∗=F(w)∗(z−w¯)PF(z) (4.16)
onH. Hence
h(z)−h(w) =(M(z)−M(w)∗)v, v
=F(w)∗(z−w¯)PF(z)v, v
=
n
j=1
(zj−w¯j)Aj(z, w)
forallz,w∈Πn,where
Aj(z, w) =PjF(z)v, F(w)v.
The Aj are clearly positive semidefinite on Πn, and hence h belongs to the Loewner
classLn. 2
5. Nevanlinnarepresentationsoftypes3, 2and1
Nevanlinna representations of type 4 have the virtue of being general for functions inLn,buttheyareundeniablycumbersome.Inthissectionweshallshowthatthereare
three simpler representation formulae, corresponding to increasingly stringent growth conditionsonh∈ Ln.
InNevanlinna’sone-variablerepresentationformulaofTheorem 1.2,
h(z) =a+bz+
1 +tz
it may be the case for a particular h ∈ P that the bz term is absent. The analogous situation intwo variables is thatthe space N inatype 4 representation maybe zero. Equivalently,inthecorrespondingHerglotz representation,theunitaryoperatorLdoes nothave1 asaneigenvalue.This suggeststhefollowingnotion.
Definition5.1.ANevanlinnarepresentationof type3 ofafunctionhonΠnconsistsofa
HilbertspaceH,a self-adjointdenselydefinedoperatorAonH,a positivedecomposition
Y ofH, a realnumberaandavectorv∈ Hsuchthat,forallz∈Πn,
h(z) =a+(1−iA)(A−zY)−1(1 +zYA)(1−iA)−1v, v
. (5.2)
Thus h has a type 3 representation if h(z) = a+M(z)v, v where M(z) is the structuredresolventofA oftype 3 correspondingto Y,asgivenbyequation (2.3).
In [5] the authors deriveda somewhat simplerrepresentation which can also be re-garded asananalogofthecaseb= 0 ofNevanlinna’s one-variableformula(5.1).
Definition5.2.ANevanlinnarepresentationof type2 ofafunctionhonΠnconsistsofa HilbertspaceH,a self-adjointdenselydefinedoperatorAonH,a positivedecomposition
Y ofH, a realnumberaandavectorα∈ Hsuchthat,forallz∈Πn
h(z) =a+(A−zY)−1α, α
. (5.3)
This meansofcoursethat,forallz∈Πn,
h(z) =a+M(z)α, α
where M(z) is the structured resolvent of A of type 2 corresponding to Y (compare equation (2.1)).
Wewish tounderstandtherelationshipbetweentype 3 andtype 2 representations.
Proposition5.3.Ifh∈ Pn hasatype2representationthenhhasatype3representation.
Conversely,ifh∈ Pn hasatype3representationasinequation(5.2)withtheadditional
property that v∈ D(A)then hhasatype 2representation.
Proof. Supposethath∈ Pn hasthetype 2 representation
h(z) =a0+
(A−zY)−1α, α
for some a0 ∈ R, positive decomposition Y and α ∈ H. We must show that h has a
representation of the form (5.2) for some a ∈ R and v ∈ H. By Proposition 2.6, it sufficesto finda∈Randv∈ D(A) suchthat
h(z) =a+(1−iA)(A−zY)−1−A(1 +A2)−1
(1 +iA)v, v
Tothisend,letC=A(1+A2)−1 andlet
a=a0+Cα, α. (5.4)
Since1+iAisinvertible onHandran(1+iA)−1⊂ D(A) wemaydefine
v= (1 +iA)−1α∈ D(A). (5.5)
Then
h(z) =a0+
(A−zY)−1α, α
=a− Cα, α+(A−zY)−1α, α
=a+(A−zY)−1−C
(1 +iA)v,(1 +iA)v
=a+(1−iA)(A−zY)−1−C
(1 +iA)v, v
asrequired.Thushhasatype 3 representation.
Conversely,lethhaveatype 3 representation(5.2)suchthatv∈ D(A),thatis
h(z) =a+M(z)v, v
where a∈ R and M is the structured resolventof A of type 3 corresponding to Y, as inequation (2.3). Since v ∈ D(A) wemay define the vector αdef= (1+iA)v ∈ H, and furthermore,byProposition 2.6,
h(z) =a+(1−iA)(A−zY)−1−C
(1 +iA)v, v
=a+(A−zY)−1−C
α, α
=a− Cα, α+(A−zY)−1α, α
=a0+
(A−zY)−1α, α
,
wherea0∈Risgivenbyequation(5.4).Thushhasarepresentationoftype 2. 2
Aspecialcaseofatype 2representationoccurswhentheconstanttermainequation
(5.3)is 0.Inonevariable,thiscorrespondstoNevanlinna’scharacterizationoftheCauchy transformsofpositivefinitemeasuresonR.Accordinglywedefineatype1representation
of h ∈ Ln to be the special case of a type 2 representation of h in which a = 0 in
equation(5.3).
Definition5.4.AnanalyticfunctionhonΠn hasaNevanlinnarepresentationoftype1 if
thereexistaHilbertspaceH,a denselydefinedself-adjointoperatorAonH,a positive decompositionY ofHandavectorα∈ Hsuchthat,forallz∈Πn,
h(z) =(A−zY)−1α, α
A representation of type 1 is obviously a representation of type 2. The following propositionisanimmediate corollaryofProposition 5.3.
Proposition 5.5. A functionh ∈ Ln has a type 1 representation if and only if h has a
type 3representation as in equation (5.2) with theadditional properties that v ∈ D(A)
and
a−A(1 +A2)−1α, α= 0.
For consistency with ourearlier terminologyfor structured resolventsand represen-tations we should haveto define a structuredresolvent of type 1 to be the sameas a structuredresolventoftype 2. Werefrainfrommakingsuchaconfusing definition.
We conclude this section bygiving examplesof the four types of Nevanlinna repre-sentationintwovariables.
Example 5.6.(1)Theformula
h(z) =− 1
z1+z2
=(0−zY)−1v, v
C,
where Y = (1 2,
1
2) andv= 1/
√
2,exhibitsarepresentationof type 1,withA= 0.
(2) Likewise
h(z) = 1− 1
z1+z2
= 1 +(0−zY)−1v, v
C
is arepresentationoftype 2.
(3) Let
h(z) =
⎧ ⎪ ⎨ ⎪ ⎩
1 1 +z1z2
z1−z2+
iz2√(1 +z12) z1z2
ifz1z2=−1
1
2(z1+z2) ifz1z2=−1
(5.7)
where wetakethebranchofthesquarerootthatisanalyticinC\[0,∞) withrangeΠ. Weclaimthath∈ P2 andthathhasthetype 3 representation
h(z) =M(z)v, vL2(R), (5.8)
where M(z) is the structured resolvent of type 3 given in Example 2.11 and v(t) = 1/π(1 +t2). Toseethis, lethbe temporarilydefinedbyequation(5.8).Sincev isan
