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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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2RONALD 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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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,
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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
14 RONALD 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
Page 16
16RONALD G. DOUGLAS, MIHAI PUTINAR AND KAI WANG
of ϕ−1
ϕ2(z1) = ϕ2(z2) and z1,z2are regular points of ϕ. Hence, if we define
2
◦ ϕ2 is just, G, a subset in that of φ−1◦ φ. Consequently, φ(z1) = φ(z2) if
ϕ1(w) = φ(z) forw = ϕ2(z),
then ϕ1is well defined outside some finite set of points in D. With a similar argument
for ϕ2, one sees that ϕ1is also a finite Blaschke product, which satisfies φ = ϕ1◦ ϕ2,
completing the proof of the lemma.
?
By the above proof, one sees that if φ is reducible, then some of the local inverses
can be analytically continued across some critical points of φ. But it is not clear that
this is a sufficient condition for φ to be reducible.
Based on the above lemma, we explain the classification for a general Blaschke prod-
uct of order four.
[11, Theorem 2.1.] Let φ be a Blaschke product of order 4. One of the following
scenarios holds.
(1) The partition of φ is {{0},{1},{2},{3}}; equivalently, φ ∼ z4.
(2) The partition of φ is {{0},{2},{1,3}}; equivalently, φ ∼ φ2
is a M¨ obius transformation with a ?= 0.
(3) The partition of φ is {{0},{1,2,3}}; equivalently, φ is not reducible.
a(z2), where φa=
a−z
1−az
All above possibility occur for some φ by the result of Sun, Zheng and Zhong in
[11].
We now classify, using purely arithmetical considerations, the possible structure for
a finite Blaschke product of order eight.
Theorem 4.2. Let φ be a Blaschke product of order 8. One of the following scenarios
holds.
(1) The partition of φ is {{0},{1},{2},{3},{4},{5},{6},{7}}; equivalently, φ ∼ z8.
(2) The partition of φ is {{0},{2},{4},{6},{1,5},{3,7}}; equivalently, φ ∼ φ2
where φa=
(3) The partition of φ is {{0},{4},{1,2,3,5,6,7}}; equivalently, φ ∼ ϕ(z2), where
ϕ is an irreducible Blaschke product of order 4.
(4) The partition of φ is one of {{0},{4},{2,6},{1,3,5,7}}, {{0},{4},{2,6},{1,3},{5,7}},
{{0},{4},{2,6},{1,5},{3,7}} or {{0},{4},{2,6},{1,7},{3,5}}; equivalently, φ ∼
ψ(ϕ2
formation with a ?= 0.
(5) The partition of φ is {{0},{2,4,6},{1,3,5,7}}; equivalently, φ ∼ ψ ◦ ϕ, where
ψ is a Blaschke product of order 2 and ϕ is an irreducible Blaschke product of order 4.
(6) The partition of φ is {{0},{1,2,3,4,5,6,7}}; equivalently, φ is not reducible.
A similar approach would work for Blaschke products of arbitrary order. However, it
seems difficult to decide whether a partition satisfying conditions (α0), (α1), (α2) and
(α3) arises from a finite Blaschke product. For example, we cannot exhibit examples
a(z4),
a−z
1−azis a M¨ obius transformation with a ?= 0.
a(z2)), where ψ is a Blaschke product of order 2 and φa=
a−z
1−azis a M¨ obius trans-
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REDUCING SUBSPACE FOR MULTIPLICATION OPERATORS OF THE BERGMAN SPACE 17
for each partition in Case (4) in Theorem 4.1, although it is likely that they exist. We
now prove Theorem 4.1 in what follows.
Proof of Theorem 4.1. By condition (α0), {0} is a singleton in the partition for
φ. Without loss of generality, suppose that G1= {0}. We list all possibilities by the
minimal number s = min{♯G2,··· ,♯Gq}, where ♯Gkis the number of elements in Gk.
Clearly s ?= 4,5,6.
(I) Case s = 1. We suppose without loss of generality that G2is also a singleton .
Subcase (A): suppose G2consists of one of the primitive element {1,3,5,7} in Z8.
Since Z8is generated by any element in {1,3,5,7}, by Conditions (α1) and (α2), each
Gkis a singleton. That is, the partition is just {{0},{1},{2},{3},{4},{5},{6},{7}}.
By [4, Lemma 8.1 and Lemma 8.3], one sees that this is equivalent to φ ∼ z8.
Subcase (B): suppose it is not subcase (A) and G2consists of 2 or 6. By Condition
(α1), the partition contains the singletons {2},{4},{6}. We list all possible partitions
as follows:
(B1)
{{0},{2},{4},{6},{1,5,3,7}};
{{0},{2},{4},{6},{1,3},{5,7}};
{{0},{2},{4},{6},{1,5},{3,7}};
{{0},{2},{4},{6},{1,7},{3,5}}.
Case (B2) is excluded by Condition (α1), since {2} + {1,3} = {3,5} is not a union of
some Gkin (B2). One can get rid of (B4) in a similar way. The remaining cases, (B1)
and (B3), satisfy (α0), (α1) and (α2). But, by a direct computation they have the
same dual partition {{0},{2},{4},{6},{1,5},{3,7}}. Using Condition (α3), we have
that {{0},{2},{4},{6},{1,5},{3,7}} is the unique choice. In this case, by Lemma
4.1, there exist a finite Blaschke product ϕ1of order 4 and a finite Blaschke product
ϕ2 of order 2 such that φ = ϕ2◦ ϕ1. Moreover, by the proof of Lemma 4.1, local
inverses for ϕ1are ρ0,ρ2,ρ4,ρ6in the family of local inverses of φ. By [4, Lemma 8.1
and Lemma 8.3], one sees that this condition is equivalent to ϕ ∼ z4. This means that
φ ∼ ψ(z4) for some Blaschke product ψ of order 2. Observe that two local inverses for
ψ are holomorphic on D, since one of them, ρ0(z) = z, is holomorphic. By [4, Lemma
8.1 and Lemma 8.3], ψ = φb◦ z2◦ φafor some M¨ obius transforms φa,φb. This implies
that φ ∼ φ2
We now consider the most complicated case in which G2= {4} is the unique singleton
other than G1. We divide it into several different subcases looking again at the minimal
number t = min{♯G3,··· ,♯Gq}. Clearly 2 ≤ t ≤ 5 and t ?= 4. So, t is 2, 3, or 5.
Subcase (C): G1= {0},G2= {4} and t = 5.
The only possibility is the partition {{0},{4},{1,2,3,5,6,7}}. By Lemma 4.1 and
the observation that ψ ∼ z2for each Blaschke product ψ of order 2, one sees that
there exists a Blaschke product ϕ of order 4 such that φ ∼ ϕ(z2). We prove that φ
is not reducible by contradiction. Otherwise, φ ∼ ϕ1◦ ϕ2, where ϕ1,ϕ1are Blaschke
products of order 2. This implies that φ ∼ ϕ1◦B for a Blaschke product B of order 4,
(B2)
(B3)
(B4)
a(z4), and a ?= 0, since it would degenerate to subcase (A) if a=0.
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18RONALD G. DOUGLAS, MIHAI PUTINAR AND KAI WANG
which leads to a contraction since by Lemma 4.1 B−1◦B forms a subgroup of order 4
in φ−1◦ φ, as desired.
Subcase (D): G1 = {0},G2 = {4} and t = 3. Then the partition consists of
G1,G2,G3,G4with ♯G3= ♯G4= 3. Considering condition (α2) and observing that 4
is the unique element other than 0 for which its inverse is itself, one sees that G−1
The following partitions are all possible choices at this point:
4
= G3.
(D1)
{{0},{4},{1,2,3},{7,6,5}};
{{0},{4},{1,2,5},{7,6,3}};
{{0},{4},{1,6,3},{7,2,5}};
{{0},{4},{1,6,5},{7,2,3}}.
The case (D1) is impossible by condition (α1), since
(D2)
(D3)
(D4)
{1,2,3} + {7,6,5} = {0,7,6,1,0,7,2,1,0}
is not a union of some subsets in (D1). One can prove similarly that (D2),(D3) and
(D4) don’t satisfy condition (α1).
Subcase (E): G1= {0},G2= {4} and t = 2.
One possibility is that the partition consists of G1,G2,G3,G4 with ♯G3 = 2 and
♯G4= 4. By Condition (α2), we have G−1
k
= Gkfor each Gk. So, the only possibilities
are:
(E1)
{{0},{4},{1,7},{2,3,5,6}};
{{0},{4},{2,6},{1,3,5,7}};
{{0},{4},{3,5},{1,2,6,7}}.
One excludes case (E1) by
(E2)
(E3)
{4} + {1,7} = {5,3},
and case (E3) by
{4} + {3,5} = {7,1}.
Another possibility is that ♯Gk = 2 for any Gk in the partition other than G1,G2.
There exist C2
6C2
4C2
2/A3
3= 15 choices:
(E4)
{{0},{4},{1,2},{3,5},{6,7}}; [(E5)]
{{0},{4},{1,2},{3,7},{5,6}}; [(E7)]
{{0},{4},{1,3},{2,6},{5,7}}; [(E9)]
{{0},{4},{1,5},{2,3},{6,7}}; [(E11)]
{{0},{4},{1,5},{2,7},{5,6}}; [(E13)]
{{0},{4},{1,6},{2,5},{3,7}}; [(E15)]
{{0},{4},{1,7},{2,3},{5,6}}; [(E17)]
{{0},{4},{1,7},{2,6},{3,5}}.
{{0},{4},{1,2},{3,6},{5,7}};
{{0},{4},{1,3},{2,5},{6,7}};
{{0},{4},{1,3},{2,7},{5,6}};
{{0},{4},{1,5},{2,6},{3,7}};
{{0},{4},{1,6},{2,3},{5,7}};
{{0},{4},{1,6},{2,7},{3,5}};
{{0},{4},{1,7},{2,5},{3,6}};
(E6)
(E8)
(E10)
(E12)
(E14)
(E16)
(E18)
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REDUCING SUBSPACE FOR MULTIPLICATION OPERATORS OF THE BERGMAN SPACE 19
One excludes most of them by the following observation: if {a,b} is included in one
of the above partitions, then one of the equations a + b = 0, a + b = 4 and a = 4 + b
holds. Indeed, by Condition (α1),
{a,b} + {a,b} = {2a,a + b,a + b,2b}
is a union of some Gk′s. If {a+b} is a singleton, then a+b = 0 or a+b = 4. Otherwise,
a + b is including in some Gksatisfying ♯Gk> 1. Noticing that each element of Gkis
included in {a,b}+{a,b}, one sees that ♯Gk≤ 3. It’s easy to verify that ♯Gk?= 3 since
we assume that the singleton {a + b} is not in the partition. So, ♯Gk= 2 and
Gk= {2a,a + b} = {a + b,2b}.
That is, 2a = 2b. This means that a = 4 + b. Furthermore, noticing that both 2a and
a + b = 2a + 4 are even in that case, one sees that Gk= {2,6}.
By this observation, all the partitions other than (E8),(E11) and (E18) are ex-
cluded. By a direct computation, one sees that (E8), (E11) and (E18) satisfy the
other conditions, too.
Moreover, the above argument shows that (E2), (E8), (E11) and (E18) are all
the possible partitions that include the sets {0},{4},{2,6}. By Lemma 4.1 and [11,
Theorem 2.1], there exists a Blaschke product ψ of order 2 and a Blaschke product ϕ
of order 4, such that φ = ψ ◦ϕ and ϕ is included in Case 2 in [11, Theorem 2.1]. This
implies that φ has the desired decomposition.
We now turn to the cases s > 1. Firstly, by Condition (α2), 4 is not included in any
Gkfor which ♯Gkis even. Otherwise, if 4 ∈ Gk, then G−1
element other than 0 for which its inverse is itself. Therefore,
k
= Gksince 4 is the unique
Gk= {4,k1,··· ,ki,8 − k1,··· ,8 − ki}
for some k1,··· ,ki. This contradicts the fact that ♯Gkis even. So, 4 / ∈ Gkif ♯Gkis
even.
Secondly, the argument used in analyzing subcase (E) is still valid. Hence, if {a,b} is
in the partition, then a+b = 0 or a = 4+b. In the latter case, {2,6} is in the partition.
Moreover, since {a,b}+{a,b} is a union of some Gk′s satisfying ♯Gk≤ 2, and 4 is not
included in any such Gk, we have that 4 ?= 2a,2b,2(a + b). Therefore, neither 2 nor
6 can be included in any Gkwhen the partition satisfies s > 1 and ♯Gk= 2. It also
implies that a + b = 0 if {a,b} is in the partition.
(II) Case s = 2.
One possibility is that the partition consists of G1,G2,G3satisfying ♯G2= 2 and
♯G3= 5. By the above observation, such partition is one of the following:
(II1)
{{0},{1,7},{2,3,4,5,6}};
{{0},{3,5},{1,2,4,6,7}}.
Obviously, none of them satisfies Condition (α1).
Another scenario is that the partition consists of G1,G2,G3,G4 satisfying ♯G2 =
♯G3= 2 and ♯G4= 3. By the above argument, G4= {2,4,6}. So, all the possibilities
are listed below:
(II2)
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20RONALD G. DOUGLAS, MIHAI PUTINAR AND KAI WANG
(II3)
{{0},{1,3},{5,7},{2,4,6}};
{{0},{1,5},{3,7},{2,4,6}};
{{0},{1,7},{3,5},{2,4,6}}.
None of them satisfies Condition (α1).
(II4)
(II5)
(III) Case s = 3.
In this case, the partition consists of G1,G2,G3satisfying ♯G2= 3 and ♯G3= 4.
By the above argument and Condition (α2), one sees that G−1
4 ∈ G2. So, the partition is one of the following:
(III1)
{{0},{1,4,7},{2,3,5,6}};
(III2)
{{0},{2,4,6},{1,3,5,7}};
(III3)
{{0},{3,4,5},{1,2,6,7}}.
Both (III1) and (III2) are excluded by Condition (α1), since {1,4,7} + {1,4,7} and
{3,4,5} + {3,4,5} are not unions of some subsets in the partitions, respectively. For
the finial possibility {{0},{2,4,6},{1,3,5,7}}, using an argument similar to the above,
one sees that it is equivalent to the condition that φ ∼ ψ ◦ ϕ, where ψ is a Blaschke
product of order 2 and ϕ is a Blaschke product of order 4, and ϕ is included in case 3
in [11, Theorem 2.1].
2
= G2, G−1
3
= G3and
(IV) Case s = 7.
The only choice is {{0},{1,2,3,4,5,6,7}}. By Lemma 4.1, φ is not reducible in this
case.
✷
We conclude with the following corollary which follows after one summarizes all the
possibilities listed above.
Corollary 4.3. Let φ be a finite Blaschke product of order 8. Then Mφhas exactly 2
nontrivial minimal reducing subspaces if and only if φ is not reducible.
It is natural to ask if this result extends to the general case. One can obtain a similar
result for order 6 by the above arithmetic way. But, the calculation for order 5 or 7
suggests that some counterexample may exist, although we can’t exhibit it. A possible
guess may be that the result holds whenever the order of φ is not prime.
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Department of Mathematics, Texas A&M University, College Station, TX 77843,
USA
E-mail address: rdouglas@math.tamu.edu
Department of Mathematics, University of California at Santa Barbara, Santa
Barbara, CA 93106, USA
E-mail address: mputinar@math.ucsb.edu
School of Mathematical Sciences, Fudan University, Shanghai, 200433, P. R. China
E-mail address: kwang@fudan.edu.cn
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