The Two-Parameter Free Unitary Segal-Bargmann Transform and its Biane-Gross-Malliavin Identification

0.68 MB·

by  Ching-Wei Ho

The Two-Parameter Free Unitary Segal-Bargmann Transform and its
Biane-Gross-Malliavin Identification
Ching-WeiHo
DepartmentofMathematics
UniversityofCalifornia,SanDiego
LaJolla,CA92093-0112
6 [email protected]
1
0
September5,2016
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p
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Abstract

2
MotivatedbyaconditionalexpectationinterpretationoftheSegal-Bargmanntransform,wederivetheinte-

] gralkernelforthelarge-N limitofthetwo-parameterSegal-Bargmann-Halltransformovertheunitarygroup
R U(N),andexploreitslimitingbehavior.Wealsoextendthenotionofcircularsystemstomoregeneralelliptic
P systems,inordertogiveanalternateconstructionofournewtwo-parameterfreeunitarySegal-Bargmann-Hall
.
h transformviaaBiane-Gross-Malliavintypetheorem.
t
a
m
Contents
[

2 1 Introduction 2
v
2
2 BackgroundandPreliminaries 6
8
1 2.1 HeatKernelAnalysisonU(N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 2.2 FreeProbability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
0
2.3 Semi-circularSystemonaFockSpace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
.
1
2.4 FreeStochasticcalculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
0
6 2.5 FreeUnitaryandFreeMultiplicativeBrownianMotions. . . . . . . . . . . . . . . . . . . . . . 13
1 2.5.1 FreeUnitaryBrownianMotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
:
v 2.5.2 FreeMultiplicativeBrownianMotion . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
i
X
3 TheConditionalExpectationRepresentation 14
r
a
4 TheIntegralTransform 16
4.1 Subordination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.2 TheTwo-ParameterHeatKernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.3 TheTwo-ParameterFreeUnitarySegal-BargmannTransform . . . . . . . . . . . . . . . . . . . 20
4.4 TheRangeoftheTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.5 LimitingBehaviorass → ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5 TheBiane-Gross-MalliavinTheorem 30
5.1 EllipticSystemsandFreeSegal-BargmannTransform . . . . . . . . . . . . . . . . . . . . . . . 30
5.2 ABiane-Gross-MalliavinTypeTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1
Acknowledgements 36
1 Introduction
In early 1960s, Segal [28, 29] and Bargmann [1, 2] introduced a unitary isomorphism from L2 to holomorphic
L2,knownastheSegal-Bargmanntransform(alsoknowninthephysicsliteratureastheBargmanntransformor
CoherentStatetransform),asamap
S : L2(Rn,ρn) → L2 (Cn,ρ2n)
t t hol t/2
where ρn is the standard Gaussian measure ( 1 )n/2exp(−1|x|2) dx on Rn and L2 (Cn,ρ ) denotes the
t 2πt 2t hol t/2
subspaceofsquareρ -integrableholomorphicfunctionsonCn. S isgivenbyconvolutionwiththeheatkernel,
t/2 t
followedbyanalyticcontinuationtoCn.
In [21], Hall generalized the Segal-Bargmann transform to any compact Lie group K; the generalization is
alsoknownastheHall’stransformortheSegal-Bargmann-Halltransform. Heconsideredtheheatkernelmeasure
ρ ofvariancetonK determinedbyanAd-invariantinnerproductonLie(K),andthecorrespondingheatkernel
t
measure µt/2 of variance 2t on the complexification KC of K; the transform, again denoted by St, is defined as
intheEuclideancaseby
Stf = (e2t∆Kf)C
where e2t∆K is the time-t heat operator on K and ( · )C means the analytic continuation from K to KC. It
is a unitary isomorphism between the Hilbert spaces L2(K,ρt) and L2hol(KC,µt). In this paper, we will be
particularly interested in the case K = U(N), the group of N ×N unitary matrices, and its complexification
KC = GL(N),thegenerallineargroupofN ×N invertiblematricesofcomplexentries.
Inanevenmoregeneralsetting,giventwopositivenumberssandtwiths > t,DriverandHallintroduced
2
in[16,22]thetwo-parameterSegal-Bargmanntransform
Ss,t : L2(K,ρs) → L2hol(KC,µs,t)
givenbythesameformulaasS butappliedtoadifferentdomainspacewhereµ isanotherheatkernelmeasure
t s,t
onKC. TheoriginaltransformSt isthesameasSt,t. Hallalsoconsideredthecaseass → ∞ands → 2t.
HavinganinfinitedimensionalversionoftheclassicalSegal-BargmanntransformforEuclideanspaces[30],
it isnatural to constructan infinitedimensional limit ofthe Segal-Bargmann transformfor compact Lie groups.
TodefinethetransformonU(N),wefixanAd-invariantinnerproductontheLiealgebrau(N) = {X ∈ M(N) :
X∗ = −X} of U(N), where M(N) is the space of all N ×N complex matrices. The most obvious approach
to the N → ∞ limit would be to use an N-independent Hilbert-Schmidt norm on all the u(N); however, M.
Gordina [18, 19] showed that this approach does not work because the target Hilbert space becomes undefined
in the limit. Indeed, Gordina showed that with the metrics normalized in this way, in the large-N limit all
nonconstantholomorphicfunctionsonGL(N)haveinfinitenormwithrespecttotheheatkernelmeasureµ .
t
Biane[9]suggestedanalternativeapproachtotheN → ∞limitoftheSegal-BargmanntransformonU(N);
instead of taking an N-independent Hilbert-Schmidt norm on all u(N), we scale the Hilbert-Schmidt norm on
u(N)byanN-dependentconstantas
N
(cid:88)
(cid:107)X(cid:107)2 = NTr(X∗X) = N |X |2. (1.1)
u(N) jk
j,k=1
WiththisN-dependentconstant,Bianecarriedoutalarge-N limitoftheLiealgebraversionofthetransform. He
consideredtheclassicalEuclideanSegal-BargmanntransformSN actingonM(N)-valuedfunctionswithnorm
t
2
(cid:107)X(cid:107)2 = NTr(X∗X), which are given by functional calculus, componentwise on the Lie algebra u(N). The
targetinnerproductspaceM(N)isequippedwithanothernorm(cid:107)X(cid:107)2 = 1Tr(X∗X). Eventhoughtheresult
N
ofapplyingSN toasingle-variablepolynomialfunctionisingeneralnotasingle-variablepolynomialfunction,
t
[9, Theorem 2] asserts that for each single-variable polynomial P, there is a unique single-variable polynomial
Pt suchthat
lim (cid:107)SNP −Pt(cid:107) = 0.
t L2(M(N),γ ;M(N))
N→∞ t/2
Recallthatγ isthevariance-tGaussianmeasureontheEuclideanspace. Biane’slimittransformmapsP toPt.
t
Inthelatersections,Bianeintroducedafreeversionof(one-parameter)Segal-Bargmanntransformbymeans
of free probability [9] as well as free stochastic calculus on a full Fock space. The underlying free probability
space is the L2 space of a semi-circular system whose construction is parallel to the classical construction of
Gaussian variables on a Boson Fock space. The range space of the free Segal-Bargmann transform is the holo-
morhpicL2 spaceofthecorrespondingcircularsystem. WeshallextendthefreeSegal-Bargmanntransformtoa
two-parameterversionwhoserangespaceistheholomorphicL2 spaceofatwo-parameterellipticsystemwhich
will be developed in Section 5.1. In the other direction of generalization, Kemp [24] studied the generalization
ofthefreeSegal-BargmanntransformondifferentFockspaces.
Bianealsoconstructedthe“large-N limitSegal-BargmanntransformonU(N)”G usingMalliavincalculus
t
techniquesandgaveaGross-Malliavinidentification[20]. G isaunitaryisomorphismbetweenanL2 spaceofa
t
measureontheunitcircleU, whichisthemultiplicativeanalogueofthesemi-circulardistribution, andarepro-
ducingkernelHilbertspaceofanalyticfunctionsonsomedomainofthecomplexplane. Inhispaper[9],Biane
didnotprovethetransformG canbeobtainedbytakingalarge-N limitfromapplyingtheSegal-Bargmanntrans-
t
form SN on U(N) to M(N)-valued functions; instead, he used the Gross-Malliavin type approach to identify
t
thepolynomialsthatshouldbetransformedtomonomials,andcomputedtheirgeneratingfunction.
TheconstructionoftheunitaryisomorphismG usesthetoolsdevelopedin[10]. Motivatedfromdevelopment
t
of free processes with free additive/multiplicative increments, Biane showed the existence of (free) Markov
transition functions for such processes. He then applied the result to the free unitary Brownian motion in [9]
to obtain a kernel which gives a free analogue of constructing the heat kernel from the Gaussian measure. The
transformG isdefinedbyintegratingtheL2 functionagainstthefreekernel.
t
Driver, Hall and Kemp [17] proved, by explicit calculation with combinatorial tools and solving partial dif-
ferential equations, that G is the direct limit of the Segal-Bargmann transform on U(N). They considered the
t
two-parameterSegal-BargmanntransformSN onU(N)actingonM(N)-valuedfunctionsandshowedthatfor
s,t
eachsingle-variablepolynomialP,eventhoughSNP istypicallynotasingle-variablepolynomial,thesequence
t
(SNP)∞ doeshaveasingle-variablepolynomiallimitG pinthefollowingsense:
s,t N=1 s,t
lim (cid:107)SNP −G P(cid:107) = 0.
s,t s,t L2(GL(N),µN ;M(N))
N→∞ s,t
They computed in [17, Theorem 1.31] that for s > t > 0, there are polynomials p(k), k = 1,2,···, such that
2 s,t
G p(k) = (·)k andthegeneratingfunctionΠ (u,z) = (cid:80) p(k)(u)zk satisfies
s,t s,t s,t k≥1 s,t
Πs,t(u,ze12(s−t)11−+zz) = (cid:16)1−uze2s11−+zz(cid:17)−1−1; (1.2)
in particular, G = G , since the generating function in s = t case defined in [9] concerning the polynomials
t,t t
whose transform under G are monomials. The main results of [17] were also proved simultaneously and inde-
t
pendentlybyCe´bronin[12],usingfreeprobabilityandcombinatoricstechniques. Thetechniques[17]and[12]
usedweredifferent;Ce´bronconsideredBrownianmotionsonU(N)andGL(N)andthefreeBrownianmotions
while Driver, Hall and Kemp did not use free probability at all. However, Ce´bron related the one-parameter
freeunitarySegal-Bargmanntransformofpolynomialstocomputingconditionalexpectations. Wewillcombine
3
Ce´bron’s and Driver, Hall and Kemp’s work to give a conditional expectation form of the two-parameter free
unitarySegal-Bargmanntransform.
GrossandMalliavin[20]showedthattheSegal-BargmanntransformoncompactLiegroupscanberecovered
fromaninfinitedimensionalversionoftheSegal-Bargmanntransformthroughtheendpointevaluationmap, by
imbedding the L2 space of the heat kernel measure on a Lie group of compact type into the L2 space of the
WienermeasureassociatedwithBrownianmotiononitsLiealgebra. Biane[9]usedafreeanalogueoftheGross-
Malliavinidentificationtodefinetheputativelarge-N limitoftheSegal-BargmanntransformonU(N). Itcanbe
recoveredfromthefreeSegal-Bargmanntransformthroughfunctionalcalculus,byimbeddingtheL2spaceofthe
measureontheunitcircleintotheL2 spaceofthefreeunitaryBrownianmotiononafreeprobabilityspace. The
freemultiplicativeBrownianmotionwasgeneralizedtothefreemultiplicative(s,t)-Brownianmotionwhichhas
beenstudiedforcouplesofyears(see,e.g.,[13,25]);itsatisfiesafreestochasticdifferentialequationwhichlooks
the same as the stochastic differential equation for (s,t)-Brownian motion on GL(N). The two-parameter free
Brownianmotionisthemainingredientoftherangespaceofthetwo-parameterfreeSegal-Bargmanntransform
whichwillbediscussedinSection5.1ofthepresentpaper.
The rest of this introduction is devoted to summary and explanation of the results of the current paper. We
consider the family of distributions ν of a free unitary Brownian motion at time t on the unit circle U whose
t
Σ-transform ftz(z) ise2t11−+zz (SeeSection2.2fordefinition). Wedenotetheinverseofft byχt,whichisanalytic
ontheunitdiskD(See[9]).
Wefirstput[12]and[17]togethertogivethefollowingproposition;seeSection3.
Proposition 1.1. Let b be the free multiplicative (s,t)-Brownian motion and u be the free unitary Brownian
s,t t
motion. Suppose that the processes u and b are free to each other. Then we have, with an abused notation
t s,t
b = b (1),
s,t s,t
G f(b ) = τ[f(b u )|b ]
s,t s,t s,t t s,t
forallLaurentpolynomialsf.
We let u and u˜ be free unitary Brownian motions which are free to each other. For s > t, the operator
t t
b (1)hasthesameholomorphicmomentsasu ,whichisstatedin[25]. Theorem2.15,whichwasprovedin
s,t s−t
[10]byBiane,asserts,withµ = ν andν = ν whereν isthedistributionofu ,thattheexistenceofaFeller
s−t t t t
MarkovkernelH = h(·,dω)onU×Usuchthat
τ[f(u u˜ )|u ] = Hf(u )
s−t t s−t s−t
foranyboundedBorelfunctionf andkernelh(ζ,dω)isdeterminedbythemomentgeneratingfunction
(cid:90) zω χ (z)ζ
s,t
h(ζ,dω) =
1−zω 1−χ (z)ζ
U s,t
whereχ = f ◦χ isananalyticfunctiononD. Itfollowsthatinthes > tcase, byProposition1.1, again
s,t s−t s
sinceu andb (1)havethesameholomorphicmoments,
s−t s,t
(cid:90)
G f(u ) = τ(f(u u˜ |u ) = Hf(u ) = f(ω)h(u ,dω)
s,t s−t s−t t s−t s−t s−t
U
which is an integral transform version of the two-parameter Segal-Bargmann transform. We will also construct
suchakernelfors ≥ t > 0. ThecomputationabovewillbegiveninmuchdetailsinSection3.
2
Having given this motivation, we then move on to establish the integral formula for the two-parameter free
unitary Segal-Bargmann transform. We are concerned with s ≥ t > 0. We first prove that χ = f ◦χ
2 s,t s−t s
is an injective conformal map from D onto its image (see Definition 4.5). Then we define a kernel k (·,dω)
s,t
whose L2 space is the same as L2(ν ); for the exact statement, see Theorem 4.6 and Proposition 4.9. And the
s
4
integralformulaforthelarge-N limitoftheSegal-BargmanntransformonU(N), calledthefreeunitarySegal-
Bargmann transform, is a unitary isomorphism from L2(ν ) to a reproducing kernel Hilbert space A defined
s s,t
foreachf ∈ L2(ν ),
s
(cid:90) |1−χ (ω)|2 1−|χ (ω)|2
G˜ f(ζ) = f(ω) s,t s,t ν (dω)
s,t (ζ −χ (ω))(ζ−1−χ¯ (ω)) 1−|χ (ω)|2 s
U s,t s,t s
for all ζ ∈ Σ where the domain Σ of the analytic function G f has a very precise description given in
s,t s,t s,t
Section 4.2. The topology of Σ depends on s only. For s < 4, Σ is simply connected while for s > 4,
s,t s,t
Σ is of conformal type as an annulus; in the complicated case s = 4, Σ itself is simply connected but the
s,t s,t
complementofΣ¯ hastwocomponents. HereisatheoremwhichsummarizesTheorem4.20andTheorem4.22:
s,t
Theorem1.2. 1. ThetransformG˜ isaunitaryisomorphismbetweentheHilbertspacesL2(ν )andthere-
s,t s
producingkernelHilbertspaceA ofanalyticfunctionsonΣ generatedbythepositive-definitesesqui-
s,t s,t
analytickernel
(cid:90) |1−χ (ω)|2 |1−χ (ω)|2 (cid:18)1−|χ (ω)|2(cid:19)2
s,t s,t s,t
K(z,ζ) = ν (dω).
(z−χ (ω))(z−1−χ¯ (ω))(ζ −χ (ω))(ζ−1−χ¯ (ω)) 1−|χ (ω)|2 s
U s,t s,t s,t s,t s
2. G˜ coincides to G , the large N-limit of the Segal-Bargmann transform on U(N); i.e. G˜ extends G
s,t s,t s,t s,t
toaunitaryisomorphismbetweenthetwoHilbertspaces.
We also compute the limit behavior of the domain Σ . If we hold t fixed and let s → ∞, the region Σ
s,t s,t
converges to an annulus with inner and outer radii e−2t and e2t; in the s = t case, if we let s = t → ∞, Σt,t is
asymptoticallyanannulusofinnerandouterradiie−2t ande2t respectively.
Wewillalsodefinethetwo-parameteranalogueofthefreeSegal-Bargmanntransformonsomefreeprobabil-
ity spaces and prove the Biane-Gross-Malliavin identification in the two-parameter setting. The L2 completion
of a semicircular system is a free analogue of the L2 space of the Gaussian measure in the classical case and
the L2 completion of the holomorphic elliptic system is the analogue of the holomorphic L2 space of a certain
anisotropic Gaussian measure, cf. [16]. The two-parameter free Segal-Bargmann transform is a unitary iso-
morphism between the two free probability spaces. The Biane-Gross-Malliavin identification is the commuting
diagram between free probability spaces: the L2 spaces of free unitary Brownian motion, free (s,t)-Brownian
motion and the L2 function spaces of the integral transform. For s > t > 0, the integral transform of the free
2
unitarySegal-BargmanntransformG canberecoveredfromthefreeSegal-BargmanntransformS through
s,t s,t
functionalcalculusonthefreeprobabilityspaces. Theidentificationispresentedinthefollowingtheorem,afull
statementisstatedasTheorem5.7.
Theorem 1.3. Let s > t > 0. Suppose u (r) is a time-rescaled free unitary Brownian motion given by the
2 s
(unique)solutionofthefreestochasticdifferentialequation
√ s
du (r) = i su (r)dx + u (r)dr.
s s r s
2
Weabusethenotationstowriteu (1)andb (1)asthefunctionalcalculusandholomorphicfunctionalcalculus
s s,t
respectively. ThenthefollowingdiagramofSegal-Bargmanntransformsandfunctionalcalculuscommute:
L2(ν ) us(1) (cid:47)(cid:47)L2(u (1),τ)
s s
G S
s,t s,t
(cid:15)(cid:15) (cid:15)(cid:15)
A (cid:47)(cid:47)L2 (b (1),τ).
s,t hol s,t
bs,t(1)
Allmapsareunitaryisomorphisms.
5
The paper is organized as follows. In section 2, we provide definitions, background and main tools for this
paper. In section 3, we will explain how we can combine [12] and [17] to give the two-parameter free unitary
Segal-Bargmanntransformintheformofconditionalexpectation. Wewillthenmakeuseoftheresulttoobtaina
simplifiedversionofanintegralversionofthetwo-parameterfreeunitarySegal-Bargmanntransform. Insection
4, we derive the integral representation for the two-parameter free unitary Segal-Bargmann transform, which is
thelarge-N limitoftheSegal-Bargmann-HalltransformonU(N),throughadirectgeneralizationoftheworkin
the previous section. In section 5, we first introduce elliptic systems which extends circular systems and define
thetwo-parameterfreeSegal-Bargmanntransform;wethenproveaversionoftheanalogueoftheBiane-Gross-
Malliavin theorem which recovers the free unitary Segal-Bargmann transform from the free Segal-Bargmann
transformbymeansoffreestochasticcalculusandfuncitonalcalculus.
2 Background and Preliminaries
2.1 HeatKernelAnalysisonU(N)
Inthissection,wegivethemainlinesofhowtoconstructtheLaplacianonU(N)andthedefinitionofthetwo-
parameter Segal-Bargmann transform on U(N). The N-dimensional unitary group U(N) is a compact matrix
Lie group with Lie algebra u(N) = {X ∈ M(N) : X∗ = −X}. The Lie algebra u(N) is equipped with the
scaledHilbert-Schmidt(real)innerproduct
(cid:104)X,Y (cid:105) = −NTr(XY). (2.1)
u(N)
Definition2.1. ForeachX ∈ u(N),theassociatedleft-invariantvectorfieldinthedirectionX isthedifferential
operator∂ : C∞(U(N),M(N)) → C∞(U(N),M(N))givenby
X
(cid:12)
(∂XF)(A) = d (cid:12)(cid:12) F(AetX)
dt(cid:12)
t=0
forallA ∈ U(N)wheneverF ∈ C∞(U(N),M(N)).
Remark2.2. Mostauthorsreferto∂ asX˜.
X
Definition2.3. Letβ beanorthonormalbasisforu(N)undertheinnerproductgivenin(2.1). TheLaplacian
N
∆ onC∞(U(N),M(N))istheoperator
U(N)
(cid:88)
∆ = ∂2
U(N) X
X∈βN
whichisindependentofthechoiceoftheorthonormalbasisβN. Fort > 0theheatoperatorise2t∆U(N) andthe
heatkernelmeasureρN ischaracterizedasthelinearfunctional
t
(cid:90) (cid:16) (cid:17)
f(U) ρNt (dU) = e2t∆U(N)f (IN)
U(N)
forallf ∈ C(U(N))whereI istheidentitymatrixinM(N).
N
TheLiegroupcomplexificationofU(N)isGL(N);inparticulargl(N,C) = u(N)⊕iu(N). Wedefinethe
Laplacian∆ tobe
GL(N)
(cid:88) (cid:88)
∆ = ∂2 + ∂2 .
GL(N) X iX
X∈βN X∈βN
6
Lets > t > 0. WedefinetheoperatorAN onC∞(GL(N),M(N))by
2 s,t
(cid:18) (cid:19)
t (cid:88) t (cid:88)
AN = s− ∂2 + ∂2 .
s,t 2 X 2 iX
X∈βN X∈βN
ThemeasureµN onGL(N)isdeterminedby
s,t
(cid:90) (cid:16) (cid:17)
f(A) µNs,t(dA) = e21ANs,tf (IN)
GL(N)
forallf ∈ C (GL(N)).
c
ObservethatANs,s = 2s∆GL(N) andAs,0 = s∆U(N);ANs,t interpolatesbetweenthetwoheatkernels.
We now give the definition of the scalar unitary Segal-Bargmann transform and boosted unitary Segal-
BargmanntransformonU(N). ThespaceM(N)isequippedwiththeinnerproduct
N
1 1 (cid:88)
(cid:104)A,B(cid:105) = Tr(B∗A) = A B¯ .
M(N) N N jk jk
j,k=1
Definition2.4. Lets > t > 0. ThescalarunitarySegal-Bargmanntransform
2
SN : L2(U(N),ρN) → L2 (GL(N),µN )
s,t s hol s,t
isdefinedbytheanalyticcontinuationofe2t∆U(N)f toanentirefunctiononGL(N);itisaunitaryisomorphism
between the two Hilbert spaces .We note that the function e2t∆U(N)f always possesses an analytic continuation
toentireGL(N)(see[15,16,23]).
ThetransformSN alsoactsonM(N)-valuedfunctionscomponentwise;weabusethenotationtodefine
s,t
SN : L2(U(N),ρN)⊗M(N) → L2 (GL(N),µN )⊗M(N)
s,t s hol s,t
whichisalsoanunitaryisomorphism. AlltensorproductsareoverC.
We will discuss the action of the boosted Segal-Bargmann transform throughout the paper; from this point
on,SN willalwaysrefertotheboostedunitarySegal-BargmannonU(N). Studyingthelarge-N limitofSN on
s,t s,t
single-variablepolynomialshelpsunderstandthelarge-N limitoftheoperatoronfunctionsgivenbyfunctional
calculus. Ingeneral,forasingle-variablepolynomialp,SN pisnotnecessarilyasingle-variablepolynomial;an
s,t
examplefrom[17]isthat,ifwetakep(u) = u2,then
(cid:20) (cid:21)
sinh(t/N)
(SN p)(Z) = e−t cosh(t/N)Z2−t ZtrZ
s,t t/N
(cid:18) (cid:19)
1
= e−t[Z2−tZtrZ]+O
N2
inwhichtrZ isinvolved. However,wehave[17,Theorem1.30]:
Theorem 2.5. Let s > t > 0, and p be a single-variable Laurent polynomial. Then there is a unique single-
2
variablepolynomialG psuchthat
s,t
(cid:18) (cid:19)
1
(cid:107)SN p−G p(cid:107)2 = O .
s,t s,t L2(GL(N),µNs,t;M(N)) N2
7
ThelimittransformG isreferredasthefreeunitarySegal-Bargmanntransform. Thefollowingtheorem[17,
s,t
Theorem1.31] showed, restricting to the space of all single-variable polynomials, G coincides to the integral
s,t
transformdefinedonanL2 spaceofameasureontheunitcircleintroducedbyBianein[9].
Theorem2.6. Lets > t > 0andlet,fork ≥ 1,p(k) bethepolynomialssuchthatG p(k) isthesingle-variable
2 s,t s,t s,t
monomialoforderk. Thenthepowerseries
Π (u,z) = (cid:88)p(k)(u)zk
s,t s,t
k≥1
convergesforallsufficientlysmall|u|and|z|,andthegeneratingfunctionΠ satisfies(1.2).
s,t
As a last remark in this section, Section 4.3 will concern the construction of the integral transform formula
ofG .
s,t
2.2 FreeProbability
Definition2.7. 1. Wecall(A,τ)aW∗-probabilityspaceifA isavonNeumannalgebraandτ isanormal,
faithfultracialstateonA. TheelementsinA arecalled(noncommuntative)randomvariables.
2. The ∗ - subalgebras A ,···A ⊆ A are called free or freely independent if, given any i ,i ,··· ,i ∈
1 n 1 2 m
{1,··· ,n}withi (cid:54)= i ,a ∈ A arecentered,thenwealsohaveτ(a a ···a ) = 0. Therandom
k k+1 ij ij i1 i2 im
variablesa ,··· ,a arefreeorfreelyindependentifthe∗-subalgebrastheygeneratearefree.
1 m
3. Foraself-adjointelementa ∈ A,thelawµofaisacompactlysupportedprobabilitymeasureonRsuch
thatwheneverf isacontinuousfunction,wehave
(cid:90)
f dµ = τ(f(a)).
R
Definition2.8(Σ-transform). LetµbeaprobabilitymeasureonC. Definethefunction
(cid:90)
ωz
ψ (z) = µ(dω)
µ
1−ωz
C
for those z with 1 (cid:54)∈ supp µ. ψ is analytic on its domain. If µ is supported in U, it is customary to restrict
z µ
ψ to the unit disk D; if µ is supported in R, it is customary to restrict ψ to the upper half-plane C . Define
µ µ +
χ = ψ /(1+ψ ). This function is injective on a neighborhood of 0 if suppµ ⊆ U and the first moment of µ
µ µ µ
isnonzero;itisinjectiveontheleft-halfplaneiC ifsuppµ ⊆ R ,cf. [7]. TheΣ-transformΣ istheanalytic
+ + µ
function
χ−1(z)
µ
Σ (z) =
µ
z
forz inaneighborhoodof0intheU-caseandforz ∈ χ (iC )intheR -case.
µ + +
Remark2.9. Thefunctionχ definedhereisusuallydenotedbyη andiscalledtheη-transformofthemeasure
µ µ
µ. Wechoosetousethenotationχherebecauseintherestofthepaper,wefollowthenotationofBianein[9].
AmeasureontheunitcircleUiscompletelydeterminedbyitsmoments;theηandΣ-transformscharacterize
the measures on U by the corresponding class of holomorphic functions on D. The corresponding class of
holomorphicfunctionsfortheη-transformisthoseanalyticselfmapsf onDsatisfying|f(z)| ≤ |z|,cf[3]; the
classoffunctionsfortheΣ-transformcanbeeasilyseenfromthedefinitionoftheΣ-transformwhichisrelated
totheη-transform.
8
For two freely independent unitary random variables x and y with laws µ and ν respectively. We define
the free multiplicative convolution µ (cid:2) ν to be the law of the unitary random variable xy. Σ-transform plays
an important role to analyze the free multiplicative convolution; it makes the free multiplicative convolution
multiplicativeinthefollowingsense:
Σµ(cid:2)ν = ΣµΣν.
Considermeasures{νt}t∈R supportedonUfort ≥ 0andsupportedonRfort ≤ 0havingΣ-transforms
t1+z
Σνt(z) = e21−z.
Writef (z) = zΣ (z)forallt ∈ R, whichisameromorphicfunctiononCwiththeonlysingularityat1. The
t νt
followingpropositionsummarizestheresultsin[4,5,6,9,33]concerningthemapsf .
t
Proposition2.10. Fort > 0,ν hasacontinuousdensityρ withrespecttothenormalizedHaarmeasureonU,
t t
theunitcircle. For0 < t < 4,itssupportistheconnectedarc
(cid:26) (cid:18) (cid:19) (cid:18) (cid:19)(cid:27)
1(cid:112) t 1(cid:112) t
suppν = eiθ : − t(4−t)−arccos 1− ≤ θ ≤ t(4−t)+arccos 1−
t
2 2 2 2
while suppν = U for t ≥ 4. The density ρ is real analytic on the interior of the arc. It is symmetric about 1,
t t
andisdeterminedbyρ (eiθ) = Reκ (eiθ)wherez = κ (eiθ)istheuniquesolution(withpositiverealpart)to
t t t
z−1
e2tz = eiθ.
z+1
Thefunctionf mapsΩ = {z ∈ D : f (z) ∈ D}ontoDconformallyandextendstoahomeomorphismfromΩ¯
t t t t
toD¯. Ω isaJordandomainand
t
(cid:18)11+− ··(cid:19)(Ωt) = (cid:40)x+iy : x > 0,(cid:12)(cid:12)(cid:12)(cid:12)xx−+11e2tx(cid:12)(cid:12)(cid:12)(cid:12) < 1,|y| < (cid:114)(x+1)2et−x−(x1−1)2etx(cid:41).
For t < 0, ν has a continuous density ρ with respect to Lebesgue measure on R . The support is the
t t +
connectedintervalsuppν = (x (t),x (t))where
t + −
(cid:112) √
2−t± t(t−4)
x±(t) = e−12 t(t−4).
2
Thedensityρ isrealanalyticontheinterval(x (t),x (t))unimodalwithpeakatitsmean1; itisdetermined
t − +
byρ (x) = 1 Iml(x)wherez = l(x)istheuniquesolutionto
t πx
z
e−t(z−12) = x.
z−1
Thefunctionf mapsΩ = {reiθ ∈ C : 0 < r < ∞,γ (r) < θ < π}whereγ (r)satisfies
t t + t t
sinγ (r) r 1
t
= −
γ (r) 1+r2−2rcos(γ (r)) t
t t
onto C conformally and extends to a homeomorphism from Ω¯ to C¯ . Ω is a Jordan domain and γ (r) is a
+ t + t t
strictly increasing function of r on the interval (z (t),1] and a strictly decreasing function of r on [1,z (t))
− +
where
(cid:112)
2+t± t(t+4)
z (t) = .
±
2
9
(cid:16) (cid:17) (cid:104) (cid:113) (cid:113) (cid:105)
Remark 2.11. When 4 > t > 0, Biane also showed in [9] that 1+· (Ω ) ∩ iR = i − 4 −1, 4 −1 .
1−· t s s
Therefore,Ω¯ ∩Uisthearc
t
 (cid:113) (cid:113) 
−i 4 −1−1 i 4 −1−1
s s
 (cid:113) , (cid:113) 
−i 4 −1+1 i 4 −1+1
s s
whichdoesnotinclude1.
Remark2.12. In[9,Proposition10],BianeprovedthatbytheHerglotzRepresentationTheorem
(cid:18) (cid:19)
1+χ (ω)
t
ν (dω) = Re dω
t
1−χ (ω)
t
whereχ = χ isdefinedonDandextendedtoahomeomorphismonD¯.
t νt
Remark 2.13. Asmentionedin[9],thefunctionf preservesinversion,forallt > 0;wecanextendχ toC\D¯
t t
sothatχ andf arestillinversetoeachother. If0 < t < 4,χ canbeanalyticallycontinuedtothecomplement
t t t
of suppν in the Riemann sphere C . For t ≥ 4, χ can be extended to C \suppν = D∪(C \D¯). χ |
t ∞ t ∞ t ∞ t D
and χ | justdifferbyaninversion. IfweputΣ = C \χ (C \suppν ), therangeofG liesinsidethe
t C∞\D¯ t ∞ t ∞ t t
HardyspaceH2(Σ ),equippedwithdifferentinnerproducts.
t
Remark 2.14. For the t < 0 case, since γ (r) is a strictly increasing function of r on the interval (z (t),1]
t −
and a strictly decreasing function of r on [1,z (t)). we have for each θ ∈ [0,γ (1)), the quadratic equation
+ t
(cid:16) (cid:17)
r2− 2cosγ (r)+ tsinγt(r) r+1 = 0hastwononnegativeroots,one< 1andtheother> 1. Soifr < 1,ris
t γt(r)
astrictlyincreasingfunctionofθ forθ ∈ [0,γ (1)].
t
Wewillcontinueusingthenotationsχ ,f ,ν ,x (t),z (t),Ω ,γ throughoutthepaper.
t t t ± ± t t
Wetrytomakesenseofthefreeconvolutionofafunctionandameasure. Wefirststateatheoremwhichwas
firstprovedin[10].
Theorem 2.15. Let (A,τ) be a W∗-probability space, B ⊆ A be a von Neumann subalgebra, and U,V ∈ A
such that U and V are unitary, with distributions µ and ν respectively. Suppose that U ∈ B and V is free with
B. ThenthereexistsaFellerMarkovkernelK = k(ζ,dω)onU×UandananalyticfunctionF definedonD
suchthat
1. foranyboundedBorelfunctionf onU,
τ(f(UV)|B) = K f(U);
2. F(z) ≤ |z|,forallz ∈ D;
3. forallz ∈ D,
(cid:90)
zω F(z)ζ
k(ζ,dω) = ;
1−zω 1−F(z)ζ
U
4. forallz ∈ D,ψµ(F(z)) = ψµ(cid:2)ν(z).
Ifµhasnonzerofirstmoment,themapF isuniquelydeterminedby(2)and(4).
TheF inTheorem2.15iscalledthesubordinationfunctionofψ withrespecttoψ . Intheclassicalcase,
νµ(cid:2)ν µ
wecanconstructfromameasureaFellerMarkovkernelbymeansofconvolution. Biane[9,10]suggestedthat,
givenameasureν onUandaboundedBorelfunctionf,withµ = δ ,K f isthefreeconvolutionofafunction
1
andameasure. Thechoiceofµ = δ canbecomparedtothekernelconstructedfrom(additive)convolutionthat
1
thekernelat0issimplytheoriginalmeasure.
10