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arXiv:1110.4920v1 [math.FA] 21 Oct 2011

REDUCING SUBSPACES FOR ANALYTIC MULTIPLIERS OF THE

BERGMAN SPACE

RONALD G. DOUGLAS, MIHAI PUTINAR AND KAI WANG

Abstract. We answer affirmatively the problem left open in [4, 8] and prove that

for a finite Blaschke product φ, the minimal reducing subspaces of the Bergman space

multiplier Mφare pairwise orthogonal and their number is equal to the number q of

connected components of the Riemann surface of φ−1◦ φ. In particular, the double

commutant {Mφ,M∗

of the minimal reducing subspaces of Mφ is also provided, along with a list of all

possible cases in degree of φ equal to eight.

φ}′is abelian of dimension q. An analytic/arithmetic description

1. Introduction

The aim of the present note is to classify the reducing subspaces of analytic Toeplitz

operators with a rational, inner symbol acting on the Bergman space of the unit disk.

While a similar study in the case of the Hardy space was completed a long time ago (see

[2, 12, 13]), investigation of the Bergman space setting was started only a few years ago.

Not surprisingly, the structure and relative position of these reducing subspaces in the

Bergman space reveal a rich geometric (Riemann surface) picture directly dependent

on the rational symbol of the Toeplitz operator.

We start by recalling a few basic facts and some terminology. The Bergman space

L2

respect to the Lebesgue measure dm on D. For a bounded holomorphic function φ on

the unit disk, the multiplication operator, Mφ: L2

a(D) is the space of holomorphic functions on D which are square-integrable with

a(D) → L2

a(D).

a(D), is defined by

Mφ(h) = φh, h ∈ L2

a(D) with symbol φ ∈ L∞(D) acts as

Tφ(h) = P(φh), h ∈ L2

The Toeplitz operator Tφon L2

a,

where P is the orthogonal projection from L2(D) to L2

ever φ is holomorphic.

An invariant subspace M for Mφis a closed subspace of L2

If, in addition, M∗

minimal reducing subspace if there is no nontrivial reducing subspace for Mφcontained

in M. The study of invariant subspaces and reducing subspaces for various classes

2010 Mathematics Subject Classification.

47B35; 30D50; 46E20.

Key words and phrases. reducing subspace, Bergman space, finite Blaschke product.

The second author was supported by NSF (DMS 1001071) and the workshop in Analysis and Prob-

ability at Texas A&M University. The third author was supported by NSFC (10731020,10801028),the

Department of Mathematics at Texas A&M University and Laboratory of Mathematics for Nonlinear

Science at Fudan University.

1

a(D). Note that Tφ= Mφwhen-

a(D) satisfying φM ⊆ M.

φM ⊆ M, we call M a reducing subspace of Mφ. We say M is a

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2 RONALD G. DOUGLAS, MIHAI PUTINAR AND KAI WANG

of linear operators has inspired much deep research and prompted many interesting

problems. Even for the multiplication operator Mz, the lattice of invariant subspaces

of L2

the lattice of reducing subspaces of Mφwas only recently made, and only in the case

of inner function symbols [4, 5, 6, 7, 8, 10, 11, 14].

Let {Mφ}′= {X ∈ L (L2

The problem of classifying the reducing subspaces of Mφis equivalent to finding the

projections in {Mφ}′. This classification problem in the case of the Hardy space was

the motivation of the highly original works by Thomson and Cowen (see [2, 12, 13]).

They used the Riemann surface of φ−1◦φ as a basis for the description of the commutant

of Mφacting on the Hardy space. Notable for our study is that inner function symbols

played a dominant role in their studies. In complete analogy, in the Bergman space

L2

analytic function f, there exists a finite Blaschke product φ such that {Mf}′= {Mφ}′.

Therefore, the structure of the reducing subspaces of the multiplier Mfon the Bergman

space of the disk is the same as that for Mφ.

Zhu showed in [14] that for each Blaschke product of order 2, there exist exactly 2

different minimal reducing subspaces of Mφ. This result also appeared in [10]. Zhu

also conjectured in [14] that Mφhas exactly n distinct minimal reducing subspaces for

a Blaschke product φ of order n. The results in [8] disproved Zhu’s conjecture, and the

authors raised a modification in which Mφwas conjecture to have at most n distinct

minimal reducing subspaces for a Blaschke product φ of order n. Some partial results

on this conjecture were obtained in [5, 8, 11]. These authors proved the finiteness

result in case n ≤ 6, each using a different method. A notable result for the general

case [8] is that there always exists a nontrivial minimal reducing subspace M, named

the ”distinguish subspace”, on which the action of Mφis unitarily equivalent to the

action of Mz on the Bergman space L2

interesting connection between the structure of the lattice of reducing subspaces of Mφ

and an isomorphism problem in abstract von Neumann algebras. The general case was

recently studied by the first author, Sun and Zheng [4] using a systematic analysis of

the local inverses of the ramified finite fibration φ−1◦φ over the disk. They proved that

the linear dimension of the commutant Aφ= {Mφ,M∗

number of connected components of the Riemann surface of φ−1◦φ. As a consequence,

one finds that the number of pairwise orthogonal reducing subspaces of Mφis finite.

In [4] the authors raised the following question, whose validity they have established

in degree n ≤ 8.

a(D) is huge and its order structure remains a mystery. Progress in understanding

a(D)) : MφX = XMφ} be the commutant algebra of Mφ.

a(D) framework, one can use essentially the same proof to show that for a ”nice”

a(D). Guo and Huang also revealed in [6] an

φ}′is finite and equal to the

Conjecture. For a Blaschke product φ of finite order, the double commutant alge-

bra Aφis abelian.

Several notable corollaries would follow once one proves the conjecture. For instance,

the commutativity of the algebra Aφimplies that, for every finite Blaschke product

φ, the minimal reducing subspaces of Mφare mutually orthogonal; in addition, their

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REDUCING SUBSPACE FOR MULTIPLICATION OPERATORS OF THE BERGMAN SPACE3

number is equal to the number q of connected components of the Riemann surface of

φ−1◦ φ.

The main result of this paper (contained in Section 2) offers an affirmative answer

to the above conjecture.

Theorem 1.1. Let φ be a finite Blaschke product of order n. Then the von Neumann

algebra Aφ= {Mφ,M∗

φ}′is commutative of dimension q, and hence Aφ∼= C ⊕ ··· ⊕ C

?

???

q

,

where q is the number of connected components of the Riemann surface of φ−1◦ φ.

The key observation for the proof is that there is an invertible holomorphic function

u such that φ = unon Ω, where Ω is a domain in D including an annulus of all

points sufficiently close to the boundary T. This implies that local inverses for φ−1◦φ

commute under composition on Ω.

It also allows us to provide an indirect description of the reducing subspaces. For

convenience, we introduce some additional notations. Following [4], there is a partition

{G1,··· ,Gq} of the local inverses for φ−1◦φ. We now define a dual partition as follows.

For two integers 0 ≤ j1,j2≤ n − 1, write j1∼ j2if

(1.1)

ζkj1=

ζkj2for any 1 ≤ i ≤ q.

?

ρk∈Gi

?

ρk∈Gi

Observing that ∼ is an equivalence relation, we partition the set {0,1,··· ,n−1} into

equivalence classes {G′

is given by the following corollary in Section 3.

1,··· ,G′

p}. Some information on the Riemann surface of φ−1◦φ

Corollary 1.2. The number of components in the dual partition is also equal to q, the

number of connected components of the Riemann surface for φ−1◦ φ.

Furthermore, we obtain the following characterization for the minimal reducing sub-

space of automorphic type in Section 3. Here O(D) denotes the space of holomorphic

functions on D.

Theorem 1.3. Let φ be a finite Blaschke product and {G′

tition for φ. Then the multiplication operator Mφ has exactly q nontrivial minimal

reducing subspaces {M1,··· ,Mq}, and for any 1 ≤ j ≤ q

Mj= {f ∈ O(D) : f|Ω∈ LΩ

where LΩ

Note the Mn−1coincides with the distinguish reducing subspace for Mφshown to exist

in [8]. This latter theorem provides a possible way to calculate the reducing subspace

if one knows the partition of the family of local inverses. The above corollary hints

that the possible partitions are very restricted.

Finally, in Section 4 we list some algebraic conditions for the partitions, which offer

an arithmetic path towards the classification of finite Blaschke products. The idea is

displayed by the classification for the Blaschke products of order 8. In a similar way

one can also explain the classifications of the Blaschke products of order 3 or 4 in

1,··· ,G′

q} be the dual par-

j},

jis a subspace of L2(Ω) with the orthogonal basis {uiu′: i + 1(modn) ∈ G′

j}.

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4RONALD G. DOUGLAS, MIHAI PUTINAR AND KAI WANG

[8, 11], which have been established by identifying the Bergman space of the disk with

the restriction of the Hardy space of the bidisk to the diagonal. We point out that

these results and examples provide some very detailed information about the branch

covering space defined by a finite Blaschke product.

2. The double commutant algebra is abelian

The notation below is borrowed from [4]. Accordingly, throughout this article φ is

a finite Blaschke product having n zeros taking multiplicity into account. The finite

set E′= φ−1(φ({β ∈ D : φ′(β) = 0})) denotes the branch points of φ, E = D\E′is

its complement in D and let Γ be a choice of curves passing through all points of E′

and a fixed point on the unit circle β0 such that D\Γ is a simply connected region

contained in E. Indeed, to be precise, one can construct Γ as follows: order E′as

{β1,β2,··· ,βs} such that k ≤ j iff Reβk≤ Reβjor Reβk= Reβjand Imβk≤ Imβj,

and set β0= Reβ1+ i?1 − (Reβ1)2. Letting Γk, 0 ≤ k ≤ s − 1 be the line segment

between βkand βk+1, we define

(2.1) Γ = ∪0≤k≤s−1Γk.

By an observation made in [4], the family of analytic local inverses {ρ0,··· ,ρn−1}

for φ−1◦ φ is well defined on D\Γ. That is, each ρjis a holomorphic function on D\Γ

which satisfies φ(ρj(z)) = φ(z) for z ∈ D\Γ. We define the equivalence relation on the

set of local inverse so that ρi∼ ρj if there exists an arc γ in E such that ρiand ρj

are analytic continuations of each other along γ. The resulting equivalence classes are

denoted {G1,··· ,Gq}. For each Gk,1 ≤ k ≤ q, define the map Ek:

(Ekf)(z) =

ρ∈Gk

?

f(ρ(z))ρ′(z), f holomorphic on D\Γ, z ∈ D\Γ.

The central result in [4] asserts that the operators {E1,··· ,Eq} can naturally be ex-

tended to bounded operators on the Bergman space L2

dent, and the double commutant algebra Aφis linearly generated by these operators;

that is,

Aφ= {Mφ,M∗

In this section we prove that the von Neumann algebra Aφis commutative.

To accomplish this, we extend the given family of analytic local inverses on D\Γ to a

larger region and prove that they commute under composition near the boundary of D.

The key observation for the proof of the following lemma is that

is a single-valued holomorphic function on C\L, where L is a curve drawn through the

zero set {a1,a2,··· ,an}. One can construct an L and verify the above assertion as fol-

lows. Notice that

−1 and ∞. By changing variables, we have for each 2 ≤ i ≤ n that

?z − ai

z − a1

a(D) which are linearly indepen-

φ}′= span{E1,··· ,Eq}.

n?(z − a1)···(z − an)

n√z + 1 is holomorphic outside any smooth simply curve connecting

n

=

n

?a1− ai

z − a1

+ 1

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REDUCING SUBSPACE FOR MULTIPLICATION OPERATORS OF THE BERGMAN SPACE5

is holomorphic outside the line segment connecting a1and ai. Therefore,

n?

(z − a1)···(z − an) = (z − a1)

n

?z − a2

z − a1···

n

?z − an

z − a1

is holomorphic outside the arc which consists of the line segments connecting a1and

aifor 2 ≤ i ≤ n. We refer the interested reader to [9, Section 55] for a more careful

argument.

Hereafter, let us set Ar= {z ∈ C : r < |z| < 1} for any 0 < r < 1, and let ζ = e

be a primitive n-th root of unity.

2iπ

n

Lemma 2.1. For a finite Blaschke product φ of order n, there exists a holomorphic

function u on a neighborhood of D\L such that φ = un, where L is an arc inside D

containing the zero set of φ. Moreover, there exists 0 < r < 1 such that Aris contained

in the image of u and u : u−1(Ar) → Aris invertible.

Proof. Suppose a1,··· ,anare the zeros of φ in D (taking multiplicity into account).

Choose an analytic branch for w =

is a single-valued holomorphic function on C\L, where L is a curve drawn through the

zero set. If we set

n√z. By [9, Section 55, p221], w =

n?(z − a1)···(z − an)

u(z) =

n?(z − a1)···(z − an)

n?(1 − a1z)···(1 − anz),

then u(z) is holomorphic on a neighborhood of D\L and un= φ.

Additionally, one sees that |u|n= |φ| on D\L and hence u(T) ⊆ T. We claim that

u(T) = T. Indeed, if u(T) ?= T, then u : T → T is homotopic to a constant map

on T. That is, there exists u(θ,t) ∈ C(T × [0,1],T) such that u(θ,0) = u(θ) and

u(θ,1) = 1. This implies that φ = un: T → T is also homotopic to the constant map

by the path t → un(·,t). If we extend each u(·,t) to be a continuous function ? u(·,t)

using [3, Theorem 1] one sees that t → Ind(T? un(·,t)) is a continuous map from [0,1]

to Z. This implies that it is a constant map, which leads to a contradiction since

−n = Ind(Mφ) = Ind(T? un(·,0)) = Ind(T? un(·,1)) = Ind(M1) = 0. Therefore, we have that

u(T) = T.

By the open mapping theorem, the image of u is an open subset of C including

T. Therefore, there exists 0 < r < 1 such that Ar ⊆ u(D\L). Now we only need

to prove that the map u : u−1(Ar) → Aris injective. In fact, for any w ∈ Ar, since

φ(u−1(ζkw)) = wnfor 0 ≤ k ≤ n − 1, we have that

?

0≤k≤n−1

Remarking that the set φ−1({wn}) includes at most n points and each set u−1({ζkw})

is nonempty, one sees that each u−1({ζkw}) is a singleton. This means that u is one to

one on u−1(Ar). Therefore, u : u−1(Ar) → Aris invertible, completing the proof.

The above lemma allows us to extend local inverses as follows. Hereafter, we denote

Ω = u−1(Ar), where Ar is the annuals appearing in Lemma 2.1. On the connected

domain Ω, define ? ρk(z) = u−1(ζku(z)) for each 0 ≤ k ≤ n − 1. Note that ? ρk is

on D, then by [3, Theorem 1] each Toeplitz operator T? un(·,t)is Fredholm. Furthermore,

u−1({ζkw}) ⊆ φ−1({wn}).

?

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6RONALD G. DOUGLAS, MIHAI PUTINAR AND KAI WANG

holomorphic and φ(? ρk(z)) = φ(z) for z ∈ Ω. This means that {? ρk}kis also the family

Matching the maps ? ρikand ρk, respectively, we obtain the family of local inverses on

Lemma 2.2. For a finite Blaschke product φ, there exists a family of local inverses for

φ−1◦φ on the domain D\Γ′, where Γ′= ∪1≤k≤s−1Γiis a proper subset of Γ appearing in

(2.1), which just consists of the set of line segments passing through all critical points

E′of φ.

of local inverses on Ω for φ−1◦ φ. It follows that ρk= ? ρikfor some ikon Ω?[D\Γ].

a larger domain Ω?[D\Γ]. Furthermore, we can prove the following lemma.

Proof. It suffices to show that the family of local inverses {ρ0,ρ1,··· ,ρn−1} can be

analytically continued across the interior point set˙Γ0= {tβ0+ (1 − t)β1: 0 < t < 1}.

To start, we prove that analytic continuation is possible when the points in˙Γ0are

close enough to the boundary T. By the continuity of u and the construction of Γ,

we can choose a number r′close to 1 such that u(Ar′) ⊂ Ar and Ar′ ∩ Γ′= ∅. For

each 0 ≤ k ≤ n − 1, let ? ρk(z) = u−1(ζku(z)) when z ∈ Ar′ (⊆ u−1(Ar)). Fix a point

{ρ0,ρ1,··· ,ρn−1} and {? ρ0, ? ρ1,··· , ? ρn−1} are local inverses of φ−1◦ φ on U. So, after

the domain Ar′ ∩ [D\Γ] = Ar′\Γ0is connected and includes U, one sees that ρi= ? ρi

[ρi∪ ? ρi](x) =

are local inverses on Ar′ ∪[D\Γ′]. We still denote them by {ρi}iwhenever no confusion

arises.

Now let S be a maximal subset of˙Γ0on which these local inverses can’t be ana-

lytically continued across. That is, {ρi}iare holomorphic on the domain D\(Γ′∪ S),

and can’t be analytically continued across each point in S. We prove S is empty by

deriving contradiction. Indeed, assume S is nonempty and let

z0 ∈ Ar′ ∩ [D\Γ], and let U be a small open disk containing z0. Notice that both

renumbering the local inverses if necessary, we can suppose that ρi= ? ρion U. Since

on this domain. Therefore, the family of analytic functions {ρi∪ ? ρi} defined as

ρi(x) if x ∈ D\Γ

? ρi(x) if x ∈ Ar′

?

s = inf{t : tβ0+ (1 − t)β1∈ S}.

Then S is contained in the line segment from z0 = sβ0+ (1 − s)β1 to β1. Since

S∩Ar′ = ∅, one sees that 0 < s and z0is inside D. This means that one can analytically

extend the local inverses across {tβ0+ (1 − t)β1: t < s}, and the process stops at z0.

But, since z0is a regular point of φ, there exists an open disk V = {z : |z − z0| < r0}

with a small r0, such that V ∩Γ′= ∅ and φ−1◦φ has n analytic branches on V . Notice

that

V ∩ [D\(Γ′∪ S)] = V \S ⊇ V \L,

where L is a line segment from the center z0to the boundary of the disk V . It follows

that V ∩ [D\(Γ′∪ S)] is a connected domain. An argument similar to that in the

preceding paragraph shows that the local inverses are holomorphic on V ∪[D\(Γ′∪S)].

By the maximality of S, we have that V ∩ S = ∅, which leads to a contradiction

since z0∈ S. Therefore, S is empty and the local inverses are holomorphic on D\Γ′,

completing the proof.

?

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REDUCING SUBSPACE FOR MULTIPLICATION OPERATORS OF THE BERGMAN SPACE7

From the proof of the above lemma one derives an intrinsic order for the local in-

verses. Specifically, we label the local inverses {ρk(z)}n−1

on Ω for 0 ≤ k ≤ n − 1. By a routine argument, we have that each ρkis invertible on

Ω, and for any pair ρk,ρk′ and z ∈ Ω, we have

ρk◦ ρk′(z) = ρk+k′modn(z).

Moreover, with little extra effort, one sees that each ρkcan also be analytically con-

tinued across the boundary T. We are now prepared to prove the main result.

k=0such that ρk(z) = u−1(ζku(z))

Theorem 2.3. Let φ be a finite Blaschke product of order n. Then the von Neumann

algebra Aφ= {Mφ,M∗

φ}′is commutative of dimension q, and hence Aφ∼= C ⊕ ··· ⊕ C

?

???

q

,

where q is the number of connected components of the Riemann surface of φ−1◦ φ.

Proof. It suffices to show that EjEi = EiEj for each 1 ≤ i,j ≤ q. Indeed, for any

0 ≤ k,k′≤ n − 1, we have that

ρk◦ ρk′(z) = ρk◦ ρk′(z) = ρk+k′modn(z),z ∈ Ω.

Therefore, for any f ∈ L2

(EiEjf)(z) =

ρ∈Gi

? ρ∈Gj

?

? ρ∈Gj

ρ∈Gi

a(D) and z ∈ Ω, we have

?

?

?

f(? ρ(ρ(z)))? ρ′(ρ(z))ρ′(z)

f(ρ(? ρ(z)))ρ′(? ρ(z))? ρ′(z) = (EjEif)(z).

a(D), completing the proof.

=

This implies that EjEi(f) = EiEj(f) for any f ∈ L2

By the final argument in the proof of [4, Theorem 8.5], the statement that Aφis

commutative is equivalent to the statement that the minimal reducing subspaces for

Mφare pairwise orthogonal. This also means that the number of distinct minimal

reducing subspaces of Mφ is equal to the dimension of Aφ. Hence, one derives the

following corollary giving the structure of the reducing subspaces.

?

Corollary 2.4. Let φ be a finite Blaschke product. Then the multiplication operator

Mφon the Bergman space L2

{M1,··· ,Mq}, and L2

nents of the Riemann surface φ−1◦ φ.

a(D) has exactly q nontrivial minimal reducing subspaces

a(D) = ⊕q

k=1Mk, where q is the number of connected compo-

3. Reducing subspaces

In order to facilitate the comprehension of the rather involved computations included

in the present section, we analyze first a simple, transparent example. If φ = zn, then

the family of local inverses is {ρk(z) = ζkz : 0 ≤ k ≤ n −1}, and we can infer without

difficulty that

Mj= span{zi: i ≥ 0, i ≡ j (modn)},1 ≤ j ≤ n

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8RONALD G. DOUGLAS, MIHAI PUTINAR AND KAI WANG

are the minimal reducing subspaces of Mzn. However, such a simple argument is not

available in the general case, so we prefer to explain the above description of the Mj

in a less direct way, as follows. Recall for φ = zn, we have that

(Ekf)(z) = f(ρk(z))ρ′

k(z) = kζkf(ζkz), 1 ≤ k ≤ n.

One verifies then that Mj is the joint eigenspace for the Ek′s corresponding to the

eigenvalues ζkj. Therefore, every Mjis a reducing subspace since the {Ek} are normal

operators and Aφ= span{E1,··· ,En}.

There is a second, more geometric description of Mjwhich emerges from this simple

example. Let Fjbe the flat bundle on D0= D\{0} with respect to the jump ζj(see

[1] for the precise definition). Roughly speaking, we cut D0along the line (0,1) in D0,

put the rank-one trivial holomorphic bundle over it, and identify the vector v on the

lower copy of (0,1) with the vector ζjv on the above copy of (0,1). Then Fjis just the

quotient space obtained from this process. One can easily see that the Fj′s are all the

flat line bundles whose pullback bundle to D0induced by the map zn: D0→ D0is the

trivial bundle. This means that each holomorphic section on Fjyields a holomorphic

function on D0by the induced composition. Let

L2

a(Fj) = {holomorphic s : D0→ Fj:

?

D0

|s|2dm < ∞},

and let Mzbe the corresponding bundle shift on L2

D0. Then the operator Uj: L2

is a unitary map, which intertwines (L2

line bundles provide a natural model for the action of Mzn on the minimal reducing

subspaces of Mzn. It is conceivable that some analogous geometric description exists

for the action of Mφon the minimal reducing subspaces in general, but, if so, we do

not know how to describe it. Thus we follow a different path below.

Returning to the general case of a finite Blaschke product φ, we will establish the

following main theorem in this section. Recall that the dual partition for φ is the

partition of the set {0,1,··· ,n − 1} for the equivalence relation defined in (1.1). We

will prove lately that the number of components in the dual partition is also equal to

q, the number of connected components of the Riemann surface for φ−1◦ φ.

Theorem 3.1. Let φ be a finite Blaschke product, and {G′

tition for φ. Then the multiplication operator Mφ has exactly q nontrivial minimal

reducing subspaces {M1,··· ,Mq}, and for any 1 ≤ j ≤ q

Mj= {f ∈ O(D) : f|Ω∈ LΩ

where Ω = u−1(Ar) is defined in Lemma 2.1, and LΩ

orthogonal basis {uiu′: i + 1(modn) ∈ G′

The remainder of this section is devoted to the proof of this theorem. We begin

with a characterization of the Mj′s in term of eigenvalues and eigenspaces of the Ek′s.

Adapting, step by step, the proof of [4, Theorem 8.5], we infer that

a(Fj). Note that |s| is well defined on

a(D)] defined by (Ujf)(z) = nzn−1f(zn)

a(Fj),Mz) and (Mj,Mzn). In this way flat

a(Fj) → Mj[⊆ L2

1,··· ,G′

q} be the dual par-

j},

jis a subspace of L2(Ω) with the

j}.

Aφ= {Mφ,M∗

φ}′= span{E1,··· ,Eq} = span{PM1,··· ,PMq},

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REDUCING SUBSPACE FOR MULTIPLICATION OPERATORS OF THE BERGMAN SPACE9

where PMkis the projection onto Mkfor 1 ≤ k ≤ q. This means that there are unique

constants {ckj,1 ≤ j,k ≤ q} such that

(3.1)

Ek=

1≤j≤q

?

ckjPMj.

On the other hand, by a dimension argument, the constant matrix [ckj] is seen to be

invertible. Since the rows of [ckj] are linearly independent, it follows that ck j1= ckj2

for each k if and only if j1= j2.

For each tuple {ckj}k, let?

Ekis a normal operator. By spectral theory,?

one sees that Mj=?

(3.2)

Mj= {f ∈ L2

We also need the following lemmas concerning the domain Ω = u−1(Ar). Let L2

be the Bergman space which consists of the holomorphic functions in L2(Ω), and let

L2

Note that since z−1∈ L2(Ω), we have that L2

the space of holomorphic functions on D.

Mj = {f ∈ L2

a(D) : Ekf = ckjf, 1 ≤ k ≤ q} be the

corresponding common eigenspace for {E1,··· ,Eq}. As shown in Theorem 2.3, each

Mj1⊥?

Mj2if j1?= j2. By the fact that

Mj⊆?

Mjfor each j, we have that?

Mj⊥Mkfor j ?= k. Noticing that L2

a(D) = ⊕kMk,

Mj. That is,

a(D) : Ekf = ckjf, 1 ≤ k ≤ q}.

a(Ω)

a,p(Ω) be the subspace of L2(Ω) which is the closure of the polynomial ring in L2(Ω).

a,p(Ω) ?= L2(Ω). Recall that O(D) denotes

Lemma 3.2. The restriction operator iΩ: L2

invertible. Furthermore, we have that L2

a(D) → L2

a(D) = {f ∈ O(D) : f|Ω∈ L2

a,p(Ω) defined by iΩ(f) = f|Ωis

a(Ω)}.

Proof. As shown in the proof of Lemma 2.2, there exists r′> 0 such that Ar′ ⊆ Ω. It’s

well known that there exists a positive constant Cr′ such that for any polynomial f

?f?L2

a(D)≤ Cr′?f?L2(Ar′).

This implies for any polynomial f that

?f?L2(D)≤ Cr′?f?L2(Ar′)≤ Cr′?f?L2(Ω)≤ Cr′?f?L2(D).

Noticing that the polynomial ring is dense in both of the two Hilbert spaces L2

L2

In addition, we have that

a(D) and

a,p(Ω), one sees that iΩis invertible.

L2

a(D) = {f ∈ O(D) : f|Ω∈ L2

It remains to show that, if f ∈ O(D) and f|Ω ∈ L2

since Ar′ ⊆ Ω, one sees that f|Ar′∈ L2

expansion for f on D. Since {zk}kare pairwise orthogonal in L2

polynomial pn=?n

Therefore, by the argument in the preceding paragraph, there exists g ∈ L2

that f|Ar′= g|Ar′. This means that f = g ∈ L2

Now we introduce operators on L2

notation, we also let Mφdenote the multiplication operator on L2

the bounded analytic symbol φ. Recall that each ρ ∈ {ρj}n−1

a,p(Ω)} ⊆ {f ∈ O(D) : f|Ω∈ L2

a(Ω), then f ∈ L2

a(Ω)}.

a(D). Indeed,

a(Ar′). Let f =?∞

k=0akzkbe the Taylor series

a(Ar′), we have that the

a(Ar′) and hence f ∈ L2

k=0akzktends to f in the norm of L2

a,p(Ar′).

a(D) such

a(D), as desired.

?

a(Ω) and L2

a,p(Ω) corresponding to {Ei}. To simplify

a(Ω) or L2

j=0is invertible on Ω.

a,p(Ω) with

Page 10

10RONALD G. DOUGLAS, MIHAI PUTINAR AND KAI WANG

Hence, the operator UΩ

operator with the inverse UΩ

EΩ

ρ: L2

a(Ω) → L2

ρ−1. Similarly, for each 1 ≤ k ≤ q, define a linear operator

a(Ω) defined by UΩ

ρ(f) = (f ◦ ρ)ρ′is a unitary

k: L2

a(Ω) → L2

a(Ω) as

EΩ

k(f) =

?

ρ∈Gk

UΩ

ρ(f) =

?

ρ∈Gk

(f ◦ ρ)ρ′, f ∈ L2

a(Ω).

Moreover, for each f ∈ L2

direct computation shows that Ek(g)|Ω= EΩ

This means that EΩ

this identity with formula (3.1) we obtain

a,p(Ω), there exists some g ∈ L2

k(f). Hence, one sees that EΩ

kis also a bounded operator on L2

a(D) such that g|Ω= f. A

k(f) ∈ L2

a,p(Ω) and iΩEk= EΩ

a,p(Ω).

kiΩ. Combining

(3.3)

EΩ

k(f) =

?

1≤j≤q

ckjiΩPMji−1

Ω(f), f ∈ L2

a,p(Ω).

Furthermore, by [4, Lemma 7.4], for each 1 ≤ k ≤ q there is an integer k−with

1 ≤ k−≤ q such that

Gk− = G−

Using an argument similar to that for [4, Lemma 7.5], we find that EΩ

fore, L2

on L2

For every 1 ≤ j ≤ q, let

MΩ

We claim that iΩPMji−1

Ω

= PMΩ

suffices to show that iΩPMji−1

that iΩPMji−1

that [ckj] is invertible, every iΩPMji−1

Ω

is a linear combination of {EΩ

that every iΩPMji−1

iΩPMji−1

We summarize the consequences of the above argument as follows.

k= {ρ−1: ρ ∈ Gk}.

k−= EΩ∗

k. There-

a,p(Ω) is a common reducing subspace of {EΩ

a,p(Ω).

k} and each EΩ

kis a normal operator

j= iΩ(Mj) = {f|Ω: f ∈ Mj}.

j. Since the range of iΩPMji−1

Ω is a projection. Indeed, a direct computation shows

Ωis an idempotent. Furthermore, combining formula (3.3) and the fact

Ω

is equal to MΩ

j, it

k}. It follows

Ω is a normal operator. Therefore, iΩPMji−1

Ω is a projection and

Ω= PMΩ

j.

Proposition 3.3. Using the notation above, L2

a,p(Ω) = ⊕q

kf = ckjf, 1 ≤ k ≤ q}.

j=1MΩ

j, and

(3.4)

MΩ

j= {f ∈ L2

a,p(D) : EΩ

In addition, one has

(3.5)

EΩ

k(f) =

?

1≤j≤q

ckjPΩ

Mj(f), f ∈ L2

a,p(Ω).

Proof. Equation (3.5) follows from formula (3.3) and the fact that iΩPMji−1

Combining this with the same argument in the beginning of the section, one sees (3.4).

Moreover, since

PMΩ

if i ?= j and

q

?

j=1

j=1

Ω= PMΩ

j.

iPMΩ

j= iΩPMiPMji−1

Ω= 0

PMΩ

j=

q

?

iΩPMji−1

Ω= I,

Page 11

REDUCING SUBSPACE FOR MULTIPLICATION OPERATORS OF THE BERGMAN SPACE 11

we have that L2

a,p(Ω) = ⊕jMΩ

j, completing the proof.

?

Since ρ1is invertible and ρn

and (UΩ

eigenvalues of UΩ

onto the eigenvector subspace

1= 1 on Ω, the operator UΩ

ρ1: L2

a(Ω) → L2

a(Ω) is unitary

i=0are possible

ρ1)n= 1. By the spectral theory for unitary operators, the {ζi}n−1

ρ1, and UΩ

ρ1=?n−1

NΩ

ρ1)j=?n−1

EΩ

i=0ζiPNΩ

i, where PNΩ

iis the projection from L2

a(Ω)

i= {f ∈ L2

i=0ζijPNΩ

a(Ω) : UΩ

ρ1(f) = ζif}.

i, andIt follows that UΩ

ρj= (UΩ

(3.6)

k(f) =

?

ρj∈Gk

n−1

?

i=0

ζijPNΩ

i(f), f ∈ L2

a(Ω).

Furthermore, we have the following lemma. Recall that u : Ω = u−1(Ar) → Ar is

invertible as shown in Lemma 2.1.

Lemma 3.4. NΩ

Proof. Since u ◦ ρ1= ζu on Ω, it is easy to check that

Uρ1(uku′) = ζiuku′, for k + 1 ≡ imodn.

That is, NΩ

show that ⊕iNΩ

orthogonal basis for L2

Define the pull-back operator Cu: L2

i= span{uku′: k ∈ Z,k + 1 ≡ imodn}.

i is contained in the eigenspace of Uρ1for the eigenvalue ζi. It remains to

i = L2

a(Ω).

a(Ar) → L2

Cuf = (f ◦ u) u′.

Since u : Ω → Aris invertible, Cuis unitary. Noticing that {zk: k ∈ Z} is a complete

orthogonal basis for L2

orthogonal basis for L2

a(Ω). In fact, we will prove that {uku′: k ∈ Z} is a complete

a(Ω) by

a(Ar), one sees that {uku′= Cu(zk) : k ∈ Z} is a complete

a(Ω), as desired.

Recall that for the partition {G1,··· ,Gq} of local inverses for φ−1◦φ, we say j1∼ j2

in the dual partition for two integers 0 ≤ j1,j2≤ n − 1, if

?

ρk∈Gi

ρk∈Gi

By this equivalence relation, the set {0,1,··· ,n − 1} is partitioned into equivalence

classes {G′

For each G′

?

ζkj1=

?

ζkj2for any 1 ≤ i ≤ q.

1,··· ,G′

jin the dual partition, let LΩ

LΩ

j=1LΩ

EΩ

kj=?

partition, c′

(3.7) yields the following result.

p}.

j= ⊕i∈G′

jNΩ

i; that is,

j= span{uiu′: i ∈ Z,i + 1(modn) ∈ G′

a(Ω). From formula (3.6)

?

1≤j≤p

j}.

Then ⊕p

(3.7)

j= L2

k(f) =c′

k jPLΩ

j(f), f ∈ L2

a(Ω),

where c′

ρi∈Gkζilfor any l ∈ G′

kj1= c′

j. By the equivalent condition for the dual

kj2for each k if and only if j1= j2. Comparing formulas (3.4) and

Page 12

12RONALD G. DOUGLAS, MIHAI PUTINAR AND KAI WANG

Proposition 3.5. For each MΩ

Proof. For each 0 ?= f ∈ MΩ

1 ≤ df ≤ p and the projection of f on LΩ

Indeed, suppose for k1?= k2, PLΩ

sees for each 1 ≤ i ≤ n that,

[PLk1+ PLk2]EΩ

Moreover, by formula (3.7),

j, there exists 1 ≤ k ≤ p such that MΩ

j⊆ ⊕kLΩ

dfis nonzero. We claim that df is unique.

k1(f) and PLΩ

j= LΩ

k∩L2

a,p(Ω).

k= L2

a(Ω), there exists at least one dfsuch that

k2(f) are nonzero. By formula (3.4), one

i(f) = cijPLk1(f) + cijPLk2(f).

[PLk1+ PLk2]EΩ

ik1= c′

i(f) = c′

ik1PLk1(f) + c′

ik2PLk2(f).

This implies that cij = c′

k1?= k2. Therefore, there exists only one integer dfsuch that PLΩ

We now prove that df is independent of f. Otherwise, there exist k1 ?= k2 and

f1,f2 ∈ Mj such that both PLΩ

proved in the preceding paragraph, we have that PLΩ

this means that both PLΩ

the uniqueness of df1+f2.

Therefore, there exists only one integer k such that PLΩ

have that cij= c′

sees that

ik2for each i. This leads to an contradiction since

df(f) ?= 0.

k1(f1) and PLΩ

k2(f2) are nonzero. By the uniqueness

k1(f2) = PLΩ

k2(f2+ f1) are nonzero, which contradicts

k2(f1) = 0. However,

k1(f1+ f2) and PLΩ

kMΩ

j?= {0}. Moreover, we

ikfor each i. Combining this fact with formulas (3.4) and (3.7), one

MΩ

j= LΩ

k∩ L2

a,p(Ω) = {f ∈ L2

a,p(D) : EΩ

if = cijf, 1 ≤ i ≤ q},

completing the proof.

?

In what follows, we will prove the converse of the above proposition. We begin with

some lemmas.

Lemma 3.6. Let f be a function holomorphic on a neighborhood of Ar. Then for any

k ∈ Z, f⊥zkin L2

Proof. Let akbe the coefficient for zkin the Laurent series expansion of f on Ar. Ob-

serve that {zk}+∞

rect computation shows that ?f,zk?L2

which leads to the desired result.

a(Ar) if and only if?

z∈Tf(z)zkdm(z) = 0.

k=−∞is a complete orthogonal basis for both of L2

a(Ar)= ak?zk?L2

a(Ar) and L2(T). A di-

a(Ar)and ?f,zk?L2(T)= ak?zk?L2(T),

?

We also need the following transformation formula.

Lemma 3.7. Let s : T → T be an invertible differentiable map. Then there exists a

constant ǫs= 1 or −1, such that for any f ∈ C(T)

?

TT

If, in addition, s is holomorphic on a neighborhood of T, then

?

TT

f(θ)dm(θ) = ǫs

?

f(s(θ))s′(θ)

is(θ)dm(θ).

f(z)dm(z) = ǫs

?

f(s(z))z s′(z)

s(z)dm(z).

Page 13

REDUCING SUBSPACE FOR MULTIPLICATION OPERATORS OF THE BERGMAN SPACE 13

Proof. It is sufficient to verify only the first equation. Indeed, the latter equation

follows from the former equation by the fact that

s′(θ) = s′(z)dz

dθ= ieiθs′(z) = iz s′(z), z ∈ T.

Without loss of generality, we can suppose that s(1) = 1. Then there exists ? s :

?

TT

Since s is invertible on T, one has that ? s : (0,2π) → (0,2π) is a monotonic function.

ferentiating the equation s(θ) = ei? s(θ), one sees that s′(θ) = iei? s(θ)? s′(θ) = is(θ)? s′(θ).

Lemma 3.8. For any integer k ≥ 0, there exists some integer i ≥ 0 such that

?zi,uku′?L2(Ω)?= 0. Therefore, PL2

Proof. We prove the statement by contradiction. Suppose that for some k ≥ 0,

?zi,uku′?L2(Ω)= 0, ∀i ≥ 0.

Since the operator Cu: L2(Ar) → L2(Ω), which appears in Lemma 3.4, is unitary, the

above equation is equivalent to

(0,2π) → (0,2π) such that s(θ) = ei? s(θ). An elementary calculus argument shows that

?

f(θ)dm(θ) =f(s(θ))|? s′(θ)|dm(θ).

Therefore, we can choose a constant ǫs= 1 or −1 such that |? s′| = ǫs? s′. Moreover, dif-

This implies that |? s′(θ)| =

a,p(Ω)NΩ

ǫss′(θ)

is(θ), completing the proof.

?

k?= {0} for all 0 ≤ k ≤ n − 1.

?(u−1)i(u−1)′,zk?L2(Ar)= 0, ∀i ≥ 0.

Using Lemma 3.6, it follows that for each integer i ≥ 0

?(u−1)i(u−1)′,zk?L2(T)=

?

T

(u−1)i(u−1)′zkdm(z) = 0.

By Lemma 3.7, Lemma 2.1 and the fact that |u(z)| = 1 for z ∈ T, we have for each

integer i ≥ 0, that

?

T

0 =zi(u−1)′◦ u(z)ukz u′(z)

u(z)dm(z) =

?

T

zi+1uk+1dm(z) = ?zi+1,uk+1?L2(T).

This means that uk+1∈ H2(T) and hence φk+1= un(k+1)∈ H2(T). Noticing that φk+1

is holomorphic on D, one sees that φk+1is a constant. This leads to a contradiction

since φ is a nontrivial Blaschke product, completing the proof.

?

Summarizing the above results, we obtain the converse of Proposition 3.5.

Proposition 3.9. For each k, there exists a unique j such that MΩ

that is,

L2

j= LΩ

k∩ L2

a,p(Ω);

a,p(Ω) = ⊕k[LΩ

k∩ L2

a,p(Ω)].

Proof. From Proposition 3.5, for each 1 ≤ j ≤ q, there exists only one 1 ≤ kj≤ p such

that MΩ

L2

j= LΩ

kj∩ L2

a,p(Ω). Hence,

a,p(Ω) = ⊕j[LΩ

kj∩ L2

a,p(Ω)].

Page 14

14RONALD G. DOUGLAS, MIHAI PUTINAR AND KAI WANG

We claim that the set {k1,··· ,kq} is just {1,··· ,p}. Indeed, if there exists k such

that 1 ≤ k ≤ p but k is not in the set {k1,··· ,kq}, then LΩ

that PL2

Lemma 3.8 we have that PL2

includes all integers between 1 and p. It follows that p = q and

k⊥ ⊕kjLΩ

k= ⊕j∈G′

kj. This means

kNΩ

a,p(Ω)LΩ

j= {0}, which leads to a contradiction, since LΩ

a,p(Ω)NΩ

j and by

j?= {0} for each j. Therefore, the set {k1,··· ,kq}

L2

a,p(Ω) = ⊕q

k=1[LΩ

k∩ L2

a,p(Ω)],

as desired.

?

In the proof of Proposition 3.9, one identifies the following intrinsic property of the

partition for a finite Blaschke product.

Corollary 3.10. The number of components in the dual partition is also equal to q,

the number of connected components of the Riemann surface for φ−1◦ φ.

Combining Lemma 3.2 with Propositions 3.5 and 3.9, we derive our main result in

this section.

Proof of Theorem 3.1. Combining Propositions 3.5 and 3.9, after renumbering if

necessary, we have for each 1 ≤ j ≤ q that,

MΩ

Noting that iΩis invertible, one sees that

j= LΩ

j∩ L2

a,p(Ω).

Mj= {f ∈ L2

a(D) : f|Ω∈ MΩ

j} = {f ∈ L2

a(D) : f|Ω∈ LΩ

j}.

Combining this formula with Lemma 3.2, we have that

Mj= {f ∈ O(D) : fΩ∈ LΩ

j},

completing the proof of the theorem.

✷

4. Arithmetics of reducing subspaces

In [8, 11], the authors obtained a classification of the structure of the finite Blaschke

product φ in case φ has order 3 or 4. In this section we show an arithmetic way towards

the classification of finite Blaschke products, displaying the details for the case of order

8.

Following [4] we define an equivalence relation among finite Blaschke products so

that φ1∼ φ2, if there exist M¨ obius transformations ϕa(z) =

with a,b ∈ D such that φ1= ϕa◦ϕ2◦ϕb. A finite Blaschke φ is called reducible if there

exist two nontrivial finite Blaschke products ϕ1,ϕ2such that φ ∼ ϕ1◦ ϕ2, and φ is

irreducible if φ is not reducible.

For a finite Blaschke product φ of order n, let G1,··· ,Gqbe the partition defined

by the family of local inverses {ρ0,··· ,ρn} for φ−1◦ φ. When no confusion arises,

we write i ∈ Gk if ρi ∈ Gk, and Gk = {i1,i2,··· ,ij} if Gk = {ρi1,ρi2,··· ,ρij}.

In view of the above notations, {G1,··· ,Gq} is a partition of the additive group

Zn= {0,1,··· ,n − 1}. One can immediately verify that, if φ1∼ φ2, then φ1,φ2yield

identical partitions.

a−z

1−azand ϕb(z) =

b−z

1−bz

Page 15

REDUCING SUBSPACE FOR MULTIPLICATION OPERATORS OF THE BERGMAN SPACE 15

The result in Corollary 3.10 hints that there should exist some internal algebraic

and combinatorial structures for the partitions arising from finite Blaschke products.

Although we don’t understand these properties completely, we list a few necessary

conditions:

(α0) {0} is a singleton in the partition, since ρ0(z) = z is holomorphic on D.

(α1) For any pair Giand Gj, there exist some Gk1,··· ,Gkmsuch that

Gi+ Gj= Gk1∪ ··· ∪ Gkm(counting multiplicities on both sides),

where ”+” is defined using the addition of Zn. (This is a consequence of the fact that

the product EiEjis a linear combination of some Ek′s).

(α2) By [4, Lemma 7.4], for each Gi= {i1,··· ,ik}, there exists j such that

Gj= G−1

i

= {n − i1,··· ,n − ik}.

(α3) By Corollary 3.10, the number of elements in the dual partition is also q.

We also need the following generalization of [4, Lemma 8.3]. Note that the additive

structure for elements in Gk′s coincides with compositions near the boundary T.

Lemma 4.1. For a finite Blaschke product φ of order n, φ is reducible if and only if

Gk1∪···∪Gkmforms a nontrivial proper subgroup of Zn, for some subset Gk1,··· ,Gkm

of the partition arising from φ.

Proof. Assume that φ is reducible. Without loss of generality, suppose that φ = ϕ1◦ϕ2

for two nontrivial finite Blaschke products ϕ1,ϕ2. Since the family of local inverses

ϕ−1

in the local inverses of φ−1◦φ, the set of the local inverses for ϕ−1

proper subgroup of φ−1◦ φ.

On the other hand, suppose that G = Gk1∪···∪Gkmis a nontrivial proper subgroup

of Znfor some Gk1,··· ,Gkm. For each Gki= {ρi1,··· ,ρij}, by [4, Thereom 3.1] there

exists a polynomial fi(w,z) of degree j such that {ρi1(z),··· ,ρij(z)} are solutions of

fi(w,z) = 0. This implies that?

pi(z),qi(z) of degree at most j. So, if we define

2◦ϕ2is a cyclic group under compositions near the boundary T, and it is contained

2◦ϕ2forms a nontrivial

ρ∈Gkiρ(z) =

pi(z)

qi(z)is a quotient of two polynomials

ϕ2(z) =

?

ρ∈G

ρ(z) =

m

?

i=1

?

ρ∈Gki

ρ(z) =

m

?

i=1

pi(z)

qi(z),

then ϕ2(z) is a rational function of degree at most ♯G; here ♯G denotes the number of

elements in G. It follows that ϕ2(z) is holomorphic outside a finite point set S of D.

Since each local inverse is bounded by 1 on D\Γ′and D\Γ′is dense in D, we have that

ϕ2is also bounded on D\S and hence it can be analytically continued across S. This

means that ϕ2 is a bounded holomorphic function on D. From a similar argument

involving local inverses, one sees that ϕ2 is also continuous on T and |ϕ2(z)| = 1

whenever z ∈ T. That implies ϕ2is a finite Blaschke product of order ♯G.

Furthermore, by the group structure of G, ϕ2(ρi(z)) = ϕ2(z) for each ρi∈ G if z is

close enough to the boundary T. Since D\Γ′is a connected domain including Ω, the

equation still holds whenever z ∈ D\Γ′. In other words, the family of local inverses