Multiplication operators on the Bergman space via analytic continuation

0.3 MB·English

by  Ronald G. Douglas

MULTIPLICATION OPERATORS ON THE BERGMAN SPACE VIA
ANALYTIC CONTINUATION
9
RONALDG.DOUGLAS,SHUNHUASUN,ANDDECHAOZHENG
0
0
2
ABSTRACT. In this paper, using the group-like property of local inverses of a finite

n Blaschke product φ, we will show that the largest C∗-algebra in the commutant of
a
the multiplicationoperatorM by φ on the Bergmanspace is finite dimensional,and
J φ
its dimensionequalsthe numberofconnectedcomponentsof the Riemannsurfaceof
3 φ−1◦φovertheunitdisk. IftheorderoftheBlaschkeproductφislessthanorequal
2
toeight,theneveryC∗-algebracontainedinthecommutantofMφisabelianandhence

] the number of minimal reducing subspaces of M equals the number of connected
φ
A componentsoftheRiemannsurfaceofφ−1◦φovertheunitdisk.
F
.
h
t
a
m 1. INTRODUCTION
[
Let D be the open unit disk in C. Let dA denote Lebesgue area measure on the
1
unit disk D, normalized so that the measure of D equals 1. The Bergman space L2 is
v a
7 the Hilbert space consisting of the analytic functions on D that are also in the space
8 L2(D,dA) of square integrable functions on D. For a bounded analytic function φ on
7
3 the unit disk, the multiplicationoperator Mφ is defined on the Bergman space L2a given
. by
1
0
M h = φh
9 φ
0 forh ∈ L2.
: a
v The classification of invariant subspaces or reducing subspaces of various operators
i
X acting on function spaces has proved to be one very rewarding research problem in
r analysis. Notonlyhastheproblemitselfturnedouttobeimportant,butalsothemethods
a
used to solve it are interesting. The classical Beurling theorem [5] gives a complete
characterizationoftheinvariantsubspacesoftheunilateralshift. Extensionsofthisidea
have led to many important works by other investigators. On the Bergman space, the
lattice of invariant subspaces of the Bergman shift M is huge and rich [4] although a
z
Beurling-typetheorem isestablishedin[2].
A reducing subspace M for an operator T on a Hilbert space H is a subspace M
of H such that TM ⊂ M and T∗M ⊂ M. A reducing subspace M of T is called
minimal if the only reducing subspaces contained in M are M and {0}. Let {M }′
φ
denote the commutant of M , which is the set of bounded operators on the Bergman
φ
space commuting with M . The problem of determining the reducing subspaces of an
φ
operator is equivalent to finding the projections in the commutant of the operator. An
1991MathematicsSubjectClassification. 47B35,30D50,46E20.
Keywordsandphrases. Reducingsubspaces,multiplicationoperators,Blaschkeproducts.
1
2 DOUGLAS,SUN,ANDZHENG
nth-orderBlaschkeproduct isthefunctionanalyticontheunitdiskD givenby
n
z −a
j
φ(z) =
1−a z
Yj=1 j
for a ∈ D. For an nth-order Blaschke product φ, since M∗ belongs to the Cowen-
j φ
Douglas class B (D), [11], [14], one can apply results from complex geometry to note
n
thatreducingsubspacescorrespondtosomesubspacesofasinglefiberwhichisisomor-
phictoCn.Thisimpliesimmediatelythattherecan’tbemorethannpairwiseorthogonal
reducing subspaces of M . However, the lattice of reducing subspaces of the bounded
φ
operatorM could stillbeinfinite
φ
LetA bethevonNeumannalgebra,definedtobetheintersectionofthecommutants
φ
{M }′ and {M∗}′. The goal of this paper is to study A for a Blaschke product φ of
φ φ φ
finite order. This is a continuation of the investigation begun in [20], [27]. In [20],
[27], one used the Hardy space of thebidisk to study multiplicationoperators on L2 by
a
bounded analyticfunctionson theunit diskD and to givecompleteclassification ofthe
reducing subspaces of multiplication operators on L2 by Blaschke products with order
a
3 or4.
In this paper, we will take a completely different approach from the one in [20],
[27]. Ononehand,themultiplicationoperatorshavedeepconnectionswiththeanalytic
properties of their symbols φ. On the other hand, those multiplication operators are
typical subnormal operators whose minimal normal extensions have a thick spectrum
and the adjoints of the multiplication operators are in the Cowen-Douglas class [11].
We will make use of two more ingredients. One is local inverses of a finite Blaschke
productandtheiranalyticcontinuationsonasubsetoftheunitdisk. Thegermsinduced
bytheselocalinverseshaveagroup-likepropertybycompositionsofgerms(fordetails
of compositions of germs, see [30]). The group-like property was used in [10], [28],
[29]instudyingthecommutantofToeplitzoperators ontheHardy space.
The other ingredient is the theory of subnormal operators [9] which, combined with
propertiesoftheCowen-Douglasclasses[11],canbeusedtoshowthatunitaryoperators
inthecommutantofthemultiplicationoperatorshaveanicerepresentation. Combining
thegroup-likepropertyoflocalinversesandthenicerepresentationofunitaryoperators,
we will obtain a symmetric and unitary matrix representation of the action of these
unitaryoperators actingon reproducingkernels.
Our main result in the paper is that the dimension of A equals the number of con-
φ
nectedcomponentsoftheRiemannsurfaceofφ−1◦φoverD. Thisresultwasobtained
forBlaschkeproductsoforder3or4in[20],[27]andsuggeststhefollowingconjecture.
Conjecture 1. For a Blaschkeproduct φ of finiteorder, thenumber of nontrivialmin-
imalreducingsubspacesofM equalsthenumberofconnected componentsof theRie-
φ
mannsurfaceofφ−1 ◦φ over D.
Our main result implies that Conjecture 1 is equivalent to that the C∗-algebra A is
φ
abelian. The conjecture is also equivalent to the statement whether or not the minimal
reducing subspaces are orthogonal. For a Blaschke product φ with order smaller than
COMMUTANT ANDREDUCINGSUBSPACES 3
or equal to 8, we will confirm the conjecture by showing that A is abelian in the last
φ
section.
This paper is motivated by Richter’s work on unitary equivalence of invariant sub-
spaces of the Bergman space [21], Stephenson’s work on hypergroups [23], [24], [25]
and Zhu’s conjecture on the number of minimal reducing subspaces of M [33]. Many
φ
ideasinthepaperare, however,inspiredbyniceworksonthecommutantofananalytic
Toeplitz operator on the Hardy space in [10], [28], [29]. C. Cowen also used the Rie-
mannsurfaceofφ−1◦φoverDtodescribethecommutantofthemultiplicationoperator
by φon theHardy space in[10].
Thomson’srepresentationoflocalinversesin[28],[29]isalsousefulintheBergman
space context. Using those local inverses as in [28], one can easily see that for an ana-
lyticand nonconstantfunctionf in theclosedunitdisk D, thereexistsafiniteBlaschke
productφ such that
{M }′ = {M }′.
f φ
This shows that the multiplication operator by a finite Blaschke product will play an
importantrolein studyingtheothermultiplicationoperators ontheBergman space.
We would like to point out that the results and arguments can carry over to the
weighted Bergman spaces, but do not work on the Hardy space. On the Hardy space,
because thespectral measure oftheminimalnormalextensionof themultiplicationop-
erator by φ is supported on the unit circle which is its essential spectrum. On the other
hand, the spectral measure of the minimal normal extension of the multiplication op-
erator by φ on the weighted Bergman space is supported on the unit disk, which is its
spectrum. The key fact is that the spectral measure is supported on the boundary of the
disk in view of the maximum principle. Although the proof doesn’t make it explicit,
we believe if points on the interior of the disk are essential with respect to the spectral
measurethentheargument goesthrough.
2. ANALYTIC CONTINUATION AND LOCAL INVERSES
Firstweintroducesomenotation. Ananalyticfunctionelementisapair(f,U),which
consists of an open disk U and an analytic function f defined on this disk. A finite
sequenceU = {(f ,U )}m isacontinuationsequenceif
j j j=1
• U ∩U isnotemptyforj = 1,··· ,m−1 and
j j+1
• f ≡ f on U ∩U , forj = 1,··· ,m−1.
j j+1 j j+1
Letγ beanarcwithparametrizationz(t),z(t)beingacontinuousfunctiononaninterval
[a,b]. Asequence{U ,··· ,U }isadmissibleoracoveringchainforγ ifeach U isan
1 m j
open disk, and if there exist increasing numbers t ,···t in [a,b] such that z(t ) ∈ U
1 m j j
forj = 1,··· ,mand
U , a ≤ t ≤ t
1 1
z(t) ∈  U ∪U , t ≤ t ≤ t
j j+1 j j+1
 U , t ≤ t ≤ b.
m m
A continuation sequence U ={(f ,U )}m is an analytic continuation along the arc
j j j=1
γ ifthesequenceU ,··· ,U isadmissiblefor γ.Each oftheelements{(f ,U )}m is
1 m j j j=1
4 DOUGLAS,SUN,ANDZHENG
ananalyticcontinuationoftheotheralongthecurveγ. Wesaythattheanalyticfunction
f on U admits a continuation to U . A famous result on analytic continuations is the
1 1 m
followingRiemann monodromytheorem[1], [18]and [30].
Theorem 2.1. Suppose Ω ⊂ C is a simply connected open set. If an analytic element,
(f,U)canbeanalyticallycontinuedalonganypathinsideΩ,thenthisanalyticfunction
element can be extended to be a single-valued holomorphic function defined on the
wholeof Ω.
Letφ bean n-th orderBlaschkeproduct. Let
E = D\[φ−1(φ({β ∈ D : φ′(β) = 0}))].
Note that D\ is finite. For an open set V ⊂ D, we define a local inverseof φ in V to be
a function f analytic in V with f(V) ⊂ D such that φ(f(z)) = φ(z) for every z in V.
That is,f isabranch ofφ−1 ◦φ defined inV.
A finitecollection,{f }, oflocalinversesin V iscompleteifforeach z in V,
i
φ−1(φ(z))∩D ⊂ {f (z)}
i
and
f (z) 6= f (z)
i j
for i 6= j. An open set V is invertible if there exists a complete collection of local
inversesinV.
Alocalinverse(f,V)admitsananalyticcontinuationalongthecurveγinEifthereis
acontinuationsequenceU = {(f ,U )}m admissibleforγ and (f ,U )equals (f,V).
j j j=1 1 1
A local inverse in V ⊂ E is admissible for φ if it admits unrestricted continuation in
E. Note that the identity function is always admissible. The set of admissible local
inverseshastheusefulpropertythatitisclosedundercomposition,whichcanbeshown
as follows. Let f and g be admissible local inverses in open discs V and W centered
at a and f(a), respectively, with f(V) ⊂ W. Let γ be a curve in E with initial point
a. Since f is admissible, it can be analytically continued along γ. There is an obvious
imagecurveγ˜ofγ underthisanalyticcontinuationalongγ. Sincegisalsoadmissible,it
can beanalyticallycontinuedalongγ˜. By refiningthecoveringchainofγ,ifnecessary,
we can assume that if V˜ is a covering disc of γ and (f˜,V˜) the corresponding function
element, then f˜(V˜) is contained in one of the covering discs of γ˜. We now compose
corresponding function elements in the analytic continuations along γ and γ˜ to obtain
an analyticcontinuationfor (g ◦f,V) alongγ.
LetV beaninvertibleopendiscandlet{f }bethefamilyofadmissiblelocalinverses
i
inV. By shrinkingV, wecan assumethateach f (V) iscontainedinan invertibleopen
i
disc W . Let {g } be the family of admissible local inverse in W . Since g ◦ f is
i ij j i ij i
admissibleandg ◦f 6= g ◦f ifj 6= k,weobservethat{g ◦f } = {f },foreach i.
ij i ik i ij i j
In particular, for each f , there exists g such that g ◦f is the identity function in V,
i ij ij i
which meansf−1 = g forsomej,and thusf−1 is admissible.
i ij i
For each z ∈ E, the function φ is one-to-one in some open neighborhood D of
zi
each point z in φ−1 ◦ φ(z) = {z ,··· ,z }. Let φ−1 ◦ φ = {ρ (z)}n be n solutions
i 1 n k k=1
φ(ρ(z)) = φ(z).Thenρ (z)islocallyanalyticandarbitrarilycontinuableinE. Assume
j
COMMUTANT ANDREDUCINGSUBSPACES 5
thatρ (z) = z. EveryopensubsetV ofE isinvertibleforφ. Then{ρ }n isthefamily
1 j j=1
of admissible local inverses in some invertible open disc V ⊂ D. For a given point
z ∈ V,labelthoselocalinversesas{ρ (z)}n onV. Ifthereisaloopγ inE atz such
0 j i=1 0
that ρ and ρ in {ρ (z)}n are mutuallyanalyticallycontinuablealong γ, we can then
j j′ i i=1
write
ρ ∼ ρ ,
j j′
anditiseasytocheckthat∼isanequivalencerelation. Usingthisequivalencerelation,
wepartition{ρ (z)}n intoequivalenceclasses
i i=1
{G ,G ,··· ,G },
i1 i2 iq
where i = 1 < i < i < ··· < i ≤ n for some integer 1 < q ≤ n and ρ is in G .
1 2 3 q ik ik
Notallofthebranchesofφ−1◦φcanbecontinuedtoadifferentbranch. Forexample,z
isasinglevaluedbranch ofφ−1◦φ. Thenq isgreaterthan1. Thuseach elementinG
ik
extendsanalyticallytotheotherelementinG ,butitdoesnotextendtoanyelementin
ik
G ifi 6= i . So
il k l
{ρ (z)}n = ∪q G . (2.1)
i i=1 k=1 ik
Thecollection{G ,G ,··· ,G } doesnot dependon thechoiceofz inE.
i1 i2 iq 0
Let k denote the reproducing kernel of the Bergman space at the point α in D. As
α
in[15], [28],wewilluselocalinversestoobtainalocal representationofan operatorT
in the commutant {M }′. The proof of the following theorem is similar to the ones in
φ
[10], [28].
Theorem 2.2. Let φ bea finiteBlaschkeproduct,U bean invertibleopen set ofE, and
t{ρ (z)}n beacompletecollectionoflocalinversesonU. ThenforeachT in{M }′,
i j=1 φ
thereareanalyticfunctions{s (α)}n onU suchthatforeach hintheBergmanspace
i i=1
L2,
a
n
Th(α) = s (α)h(ρ (α)),
i i
Xi=1
n
T∗k = s (α)k
α i ρi(α)
Xi=1
for each α in U. Moreover, these functions {s (α)}n admit unrestricted continuation
i i=1
in E.
Proof. Since T commutes with M , the adjoint T∗ commutes with M∗. Thus T∗ com-
φ φ
mutes with M∗ for each α in the invertible set V. So the kernel of M∗ is
φ−φ(α) φ−φ(α)
invariantforT∗.NotethatthekernelofM∗ isthefinitedimensionalspacespanned
φ−φ(α)
by {k }n . Hence foreach α inU, there isa sequence{s (α)}n ofcomplexnum-
ρi(α) i=1 i i=1
bers such that
n
T∗k = s (α)k .
α i ρi(α)
Xi=1
6 DOUGLAS,SUN,ANDZHENG
Thusforeach hinL2,wehave
a
Th(α) = hTh,k i
α
= hh,T∗k i
α
n
= hh, s (α)k i
i ρi(α)
Xi=1
n
= s (α)hh,k i
i ρi(α)
Xi=1
n
= s (α)h(ρ (α)).
i i
Xi=1
Tofinishtheproofweneedtoshowthat{s (z)}n areanalyticinU. Todoso,foreach
i i=1
i, define
n
P (α,z) = (z −ρ (α))
i j
Y
j6=i
for z in D and α in U. Thus {P (α,z)} is a family of functions analytic in z on D and
i
analyticin αon U. An easy calculationgivesthat foreach α inU
n
hP (α,.),T∗k i = hP (α,.), s (α)k i
i α i j ρi(α)
Xj=1
n
= s (α)hP (α,.),k i
j i ρi(α)
Xj=1
= s (α)P (α,ρ (α))
i i i
n
= (ρ (α)−ρ (α)).
i j
Y
j6=i
Thus
hP (α,.),T∗k i
i α
s (α) =
i n (ρ (α)−ρ (α))
j6=i i j
for α in U and hence s is analytic Qin U. Noting that {ρ (z)}n admit unrestricted
i i j=1
continuationinE, weconcludethatthefunctions{s (α)}n admitunrestrictedcontin-
i i=1
uationin E to completetheproof.
3. RIEMANN SURFACES φ−1 ◦φ OVER D
Let φ = P(z) be an n-th order Blaschke product of where P(z) and Q(z) are two
Q(z)
coprimepolynomialsof degreeless than or equal to n. In thissection wewill studythe
RiemannsurfacefortheBlaschkeproductφ−1◦φoverD. Inparticular,itwasshownin
[27]howthenumberoftheconnectedcomponentsoftheRiemannsurfaceφ−1◦φover
D isrelated to thezeros ofφ forthefourthorderBlashckeproduct φ. Let
f(w,z) = P(w)Q(z)−P(z)Q(w).
COMMUTANT ANDREDUCINGSUBSPACES 7
Thenf(w,z)isapolynomialofw withdegreenanditscoefficientsarepolynomialsof
z with degree n. For each z ∈ D, f(w,z) = 0 has exactly n solutions in D counting
multiplicity. An algebraic function is a function w = g(z) defined for values z in D by
an equationf(w,z) = 0.
LetC denotethesetofthecriticalpointsofφ inD and
F = φ−1 ◦φ(C) = {z ,··· ,z }.
1 m
ThenF isafinitesetandiscalledthesetofbranchpointsofφ,andφ−1◦φ = {ρ (z)}n
k k=1
isann-branchedanalyticfunctiondefinedandarbitrarilycontinuableinD\F. Notallof
thebranchesofφ−1◦φcanbecontinuedtoadifferentbranch,forexamplez isasingle
valued branch of φ−1 ◦φ. The Riemann surface S for φ−1 ◦φ over D is an n-sheeted
φ
cover of D with at most n(n − 1) branch points, and it is not connected. We denote a
pointofS lyingoverD\F by(ρ(α),α),whereαisinD\F andρisabranchofφ−1◦φ
φ
defined in aneighborhoodofα.
Visualizationof Riemann surfaces is complicated by the fact that they are embedded
in C2, a four-dimensional real space. One aid to constructing and visualizing them is a
methodknownas“cutandpaste”. HerewepresentonlydetailsonhowtoconstructS .
f
For general cases, see [3], [6], [17]. We begin with n copies of the unit disk D, called
sheets. The sheets are labeled D ,··· ,D and stacked up over D. Then {z ,··· ,z }
1 n 1 m
are the branch points. Suppose Γ is a curve drawn through those branch points and a
fixedpointontheunitcirclesothatD\Γisasimplyconnected region. By theRiemann
monodromytheorem, n distinctfunction elements ρ (z), k = 1,··· ,n of thealgebraic
k
equation
f(w,z) = 0
can be extended to be a single-valued holomorphic functions defined over the whole
of D\Γ. We denote these extended functions still by ρ (z). We may assume that Γ
j
consists of line segments l to connect z to z . The sheets D are cut open along
k k k+1 j
those line segments l . Then various sheets are glued to others along opposite edges of
k
cuts. With the point in the k-th sheet over a value z in D\Γ we associate the pair of
values (ρ (z),z). In this way a one-to-one correspondence is set up between the points
k
in S over D\Γ and the pair of points on the n sheets over D\Γ. In order to make the
f
correspondence continuous along the cuts exclusive of their ends, let two regions R
1
and R be defined in a neighborhood of each cut l . On each of the n sheets, in the
2 i
region formed by R , R and the cut l between them exclusive of its ends, the values
1 2 i
of the algebraic function w = g(z) form again n distinct holomorphic functions ρ (z)
k
(k = 1,··· ,n),andthesecanbenumberedsothatg (z) = ρ (z)inR .IntheregionR
l l 1 2
the functions g (z) are the same functions in the set {ρ (z)} but possibly in a different
k k
order. We join the edge of the cut bounding R in the k-th sheet to the edge bounding
1
R in the l-th sheet, where l is so determined that g (z) = ρ (z) in R . The continuous
2 k l 2
Riemann surfacesoformed has thepropertythat pointsintheRiemann surfaceS over
f
non-branch points D\{z ,··· ,z } are in one-to-one continuous correspondence with
1 m
thenonsingularpoints (w,z) which satisfies theequation f(w,z) = 0. We not onlyget
amanifold;thatis,theseidentificationsarecontinuousbuttheRiemannsurfacealsohas
an analyticstructureorthematch upsare analytic.
8 DOUGLAS,SUN,ANDZHENG
We need to use the number of connected components of the Riemann surface S
φ
in the last two sections. By the unique factorization theorem for the ring C[z,w] of
polynomialsinz and w,wecan factor
q
f(w,z) = p (w,z)nj,
j
Yj=1
wherep (w,z),··· ,p (w,z)areirreduciblepolynomials. Bochner’sTheorem[31]says
1 q
thatφ has finitelymanycriticalpointsintheunitdiskD. Thuswehave
q
f(w,z) = p (w,z).
j
Yj=1
The following theorem implies that the number of connected components equals the
numberofirreduciblefactorsf(w,z).ThisresultholdsforRiemannsurfacesovercom-
plexplane(cf. [6], page78 and[17], page374).
Theorem3.1. Letφ(z)beann-thorderBlaschkeproductandf(w,z) = q p (w,z).
j=1 j
Suppose that p(w,z) is one of factors of f(w,z). Then the Riemann surfQace S is con-
p
nected if and only if p(w,z) is irreducible. Hence q equals the number of connected
componentsoftheRiemannsurfaceS = S .
φ f
Proof. Let {z }m be the branch points of p(w,z) = 0 in D. Bochner’s Theorem [31]
j j=1
says that those points {z }m are contained in a compact subset of D. Suppose that
j j=1
p(w,z) is irreducible. If the Riemann surface S is not connected, let {ρ (z)}np be
p k k=1
n distinct branches of p(w,z) = 0 over D\Γ. Then {ρ (z)}np are also roots of the
p k k=1
equation
φ(w)−φ(z) = 0.
Assuming that S is not connected, we will derive a contradiction. Suppose that one
p
connected component of S is made up of the sheets corresponding to {ρ ,··· ,ρ }
p 1 n1
(n < n ). Let σ (x ,··· ,x ) be elementary symmetric functions of variables x ,
1 p s 1 n1 1
··· ,x with degrees:
n1
σ (x ,··· ,x ) = x x ···x .
s 1 n1 j1 j2 js
X
1≤j1<j2<···<js≤n1
SincethecontinuationofanypathinD\F onlyleadstoapermutationin{ρ (z),··· ,ρ (z)},
1 n1
every σ (z) = σ (ρ (z),··· ,ρ (z)) is unchangedundersuch apermutationand hence
s s 1 n1
is a holomorphic function well-defined on D\{z }m and analytically extends on a
j j=1
neighborhoodoftheunitdiskalthoughρ (z) isdefined onlyon D\Γ.
j
Notethatρ (z)isinD. Thusσ (ρ (z),··· ,ρ (z))isboundedonD\{z }m . Bythe
j s 1 n1 j j=1
Riemann removable singularities theorem, σ (ρ (z),··· ,ρ (z)) extends analytically
s 1 n1
on tD for some t > 1. Now we extend σ (ρ (z),··· ,ρ (z)) to the complex plane C.
s 1 n1
Foreach z ∈ C\D,define
1 1
f (z) = σ ( ,··· , ).
s s
ρ (1) ρ (1)
1 z¯ n1 z¯
COMMUTANT ANDREDUCINGSUBSPACES 9
By Theorem 11.1 on page 25 [6], near an ordinary point z = a, each function ρ (z)
j
has a power series of z − a. By Lemma 13.1 on page 29 [6], each function ρ (z) has
j
a Laurent series of a fractional power of (z −a) but the number of terms with negative
exponents must be finite. Thus f (z) is a meromorphic function in C\D and the point
s
at infinity is a pole of each f (z). Moreover, f (z) is analytic in a neighborhood of the
s s
unitcircle. So f (z) isanalyticin tD\rDfor0 < r < 1 < t.
s
Nextweshowthaton theunitcircleexcept foronepoint,
1
= ρ (z)
i
ρ (1)
i z¯
for each i. To do this, for each ρ ∈ {ρ }n , noting thatφ is analyticon a neighborhood
i i=1
V of the closure of the unit disk and ρ extends analytically on the unit circle minus Γ,
wehavethatforeach w ∈ V with|z| = 1,
|ρ(z)| = 1.
To provetheabovefact, wefollowan argumentin [28]. If |ρ(z)| < 1, as
φ(ρ(z)) = φ(z)
and z is not a branch point of φ, then ρ(z) is in D\{z }m . Thus ρ−1 is a local inverse
j j=1
near ρ(z) but its range is not contained in the unit disk D. So ρ−1 is not admissibleand
neitherisρ. Hence |ρ(z)| = 1.Thismeans
1
ρ(z) =
ρ(1)
z¯
for|z| = 1.
Thus
σ (ρ (z),··· ,ρ (z)) = f (z).
s 1 n1 s
forz ontheunitcircleexcept foronepoint. So
σ (ρ (z),··· ,ρ (z)) = f (z)
s 1 n1 s
in aneighborhoodoftheunitcircle. Define
σ (ρ (z),··· ,ρ (z)) z ∈ D¯
F (z) = { s 1 n1
s f (z) z ∈ C\D.
s
Thus F (z) is a meromorphic function in C and the point at infinity is a pole of each
s
F (z). Hence F (z) is arationalfunctionofz andso isσ (ρ (z),··· ,ρ (z)) inD.
s s s 1 n1
Nowconsiderthepolynomial
n1
f (w,z) = wn1 −σ (z)wn1−1 +···+(−1)n1σ (z) = (w −ρ (z))
1 1 n1 j
Yj=1
whosecoefficientsare rationalfunctionsofz. Thus
p(w,z) = f (w,z)f (w,z)
1 2
10 DOUGLAS,SUN,ANDZHENG
for another polynomial f (w,z). This implies that p(w,z) is reducible, which con-
2
tradicts the assumption that p(w,z) is irreducible. Hence the Riemann surface S is
p
connected.
If p(w,z) is reducible, notingthat every root ρ(z) ofp(w,z) is also aroot of φ(w)−
φ(z), by Bochner’s Theorem [31], we see that p(w,z) does not have multiple roots for
z ∈ D\{z }m ,wecan factor
j j=1
p(w,z) = p (w,z)···p (w,z)
1 q1
forsomeirreduciblepolynomialsp (w,z) withdegreek .
l l
Let{ρ (z),··· ,ρ (z)} berootsoftheequationp (w,z) = 0. Thus
l1 lkl l
p (ρ (z),z) = 0.
l lj
From the identity theorem of analytic functions, each analytic continuationρˆ (z) must
lj
stillsatisfytheequation
p (ρˆ (z),z) = 0,
l lj
and so ρˆ (z) must be in {ρ }kl . Since p is irreducible, by the above argument, we
lj lj j=1 l
seethatthecontinuationsoftherootsρ (z),··· ,ρ (z)ofp (w,z)arealwaysrootsof
l1 lkl l
p (w,z). Hence crossing a cut permutes the set of roots, and the Riemann surface S
l pl
is connected. This gives that S has q connected components, one for each of factors
p 1
p ,··· ,p . In particular, S has q connectedcomponents. Thiscompletestheproof.
1 q1 f
4. REPRESENTATION OF UNITARY OPERATORS
In this section, we will obtain a better representation of a unitary operator in the
commutant {M }′ on E\Γ. It will be defined using the orientation {{ρ }n ,U} of a
φ i i=1
complete collection of local inverses {ρ } on U for a small invertible open set U of E.
i
Thetheoryofsubnormaloperators[9] playsalsoan importantrolein thissection.
Next we need to order the set {ρ }n globally over a simply connected subset of E.
j j=1
To do this, take an invertible small open set U of E such that the intersection of ρ (U)
j
and ρ (U) is empty forj 6= k. We can always do so by shrinkingU sufficiently. In this
k
section,wefix thesmallinvertibleopen setU. Welabel {ρ (z)}n as
j j=1
{ρ (z),ρ (z),··· ,ρ (z)}
1 2 n
and assume ρ (z) = z. Take a curve Γ through the finite set F and connecting a point
1
on the unit circle so that E\Γ is simplyconnected and disjointfrom theset ∪n ρ (U).
j=1 j
Theorem 2.1 (the Riemann Monodromy theorem) gives that each of {ρ (z)}n has a
j j=1
uniquelyanalyticcontinuationonE\Γandhencewecanvieweachof{ρ (z)}n asan
j j=1
analyticfunctionon E\Γ satisfying
φ(ρ (z)) = φ(z),
j
forz ∈ E\Γandj = 1,··· ,n.Weretainthesamelabels{ρ (z)}n as{ρ (z),ρ (z),··· ,ρ (z)}
j j=1 1 2 n
ateverypointinE\Γ. Theorientationisdenotedby{{ρ (z)}n ,U}. Thenthecompo-
j j=1
sitionρ ◦ρ (z) makes senseon U. As wepointedout before, {ρ }n has agroup-like
k l j j=1
property undercompositiononU, thatis,
ρ ◦ρ (z) = ρ (z), (4.1)
k i πi(k)