Reducing subspaces for analytic multipliers of the Bergman space
ABSTRACT We answer affirmatively the problem left open in \cite{DSZ,GSZZ} and prove
that for a finite Blaschke product $\phi$, the minimal reducing subspaces of
the Bergman space multiplier $M_\phi$ are pairwise orthogonal and their number
is equal to the number $q$ of connected components of the Riemann surface of
$\phi^{1}\circ \phi$. In particular, the double commutant
$\{M_\phi,M_\phi^\ast\}'$ is abelian of dimension $q$. An analytic/arithmetic
description of the minimal reducing subspaces of $M_\phi$ is also provided,
along with a list of all possible cases in degree of $\phi$ equal to eight.

Article: Cowen's class and Thomson's class
[Show abstract] [Hide abstract]
ABSTRACT: In studying commutants of analytic Toeplitz operators, Thomson proved a remarkable theorem which states that under a mild condition, the commutant of an analytic Toeplitz operator is equal to that of Toeplitz operator defined by a finite Blaschke product. Cowen gave an significant improvement of Thosom's result. In this paper, we will present examples in Cowen's class which does not lie in Thomson's class.12/2013;  SourceAvailable from: export.arxiv.org[Show abstract] [Hide abstract]
ABSTRACT: Given an ntuple of multiplication operators on the Bergman space of a bounded pseudoconvex domain in C^n, we study the algebra of their commutants. In particular, we give a geometric description of the maximal C*subalgebra of this algebra.10/2013;  SourceAvailable from: Jaydeb Sarkar[Show abstract] [Hide abstract]
ABSTRACT: In this article we give an introduction of Hilbert modules over function algebras and a survey of some recent developments. We present the theory of Hilbert modules as combination of commutative algebra, complex geometry and the geometry of Hilbert spaces and its applications to the theory of $n$tuples ($n \geq 1$) of commuting operators.08/2013;
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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 squareintegrable 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
Page 3
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) : fΩ∈ 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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4 RONALD 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 singlevalued 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
Page 5
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 nth 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 singlevalued 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 un= 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}).
?