Shuaibing Luo, Hunan University Joint work with Caixing Gu ... · ReducingsubspacesoftheDirichletspace Shuaibing Luo, Hunan University Joint work with Caixing Gu (California Polytechnic

Reducing subspaces of the Dirichlet space

Shuaibing Luo, Hunan University

Joint work with Caixing Gu (California Polytechnic State University)and Jie Xiao (Memorial University)

Knoxville

Shuaibing Luo, Hunan University Joint work with Caixing Gu (California Polytechnic State University) and Jie Xiao (Memorial University)Reducing subspaces of the Dirichlet space Knoxville 1 / 15

Notations

Open unit disc: D = {z ∈ C : |z | < 1}


Notations


Hardy space: H2(D) = {f ∈ Hol(D) : ‖f ‖2H2 =

∑∞

n=0 |an|2 <

∞, f (z) =∑

∞

n=0 anzn}

Bergman space:L2a(D) = {f ∈ Hol(D) : ‖f ‖2

L2a(D)=

∫D|f (z)|2dA(z) < ∞}


Notations


Hardy space: H2(D) = {f ∈ Hol(D) : ‖f ‖2H2 =

∑∞

n=0 |an|2 <

∞, f (z) =∑

∞

n=0 anzn}

Bergman space:L2a(D) = {f ∈ Hol(D) : ‖f ‖2

L2a(D)=

∫D|f (z)|2dA(z) < ∞}

Dirichlet space: D = {f ∈ Hol(D) :∫D|f ′|2dA < ∞}

norm: ‖f ‖2D = ‖f ‖2H2 +

∫D|f ′|2dA


Background on reducing subspaces

Finite Blaschke productMobius transform: ϕλ(z) =

λ−z

1−λz, λ ∈ D




λ−z

1−λz, λ ∈ D

Finite Blaschke product of order n: φ = ϕλ1· · ·ϕλn

, λi ∈ D




λ−z

1−λz, λ ∈ D


, λi ∈ D

Reducing subspace: Let H = H2, L2a or D, T ∈ B(H). A closedsubspace M is called a reducing subspace of T if M is invariant forboth T and T ∗




λ−z

1−λz, λ ∈ D


, λi ∈ D

Reducing subspace: Let H = H2, L2a or D, T ∈ B(H). A closedsubspace M is called a reducing subspace of T if M is invariant forboth T and T ∗

On H2, the reducing subspaces of Mφ are in one-to-onecorrespondence with the closed subspaces of H2 ⊖ φH2


On L2a: when n = 2, in 1998, S. L. Sun, Y. J. Wang showed that Mφ

has exact 2 minimal reducing subspaces on L2a. In 2000, Zhu provedthis result using a different method; Zhu conjectured that for a finiteBlaschke product φ of order n, there are exactly n distinct minimalreducing subspaces of Mφ on L2a.




Douglas, Guo, Hu, Huang, Putinar, S. L. Sun, S. H. Sun, Y. J. Wang,K. Wang, Xu, Yu, Zheng, Zhong, etc. considered Zhu’s conjecture,they found that Zhu’s conjecture does not hold in general, and it ismodified as follows: Mφ has at most n distinct minimal reducingsubspaces on L2a.




Douglas, Guo, Hu, Huang, Putinar, S. L. Sun, S. H. Sun, Y. J. Wang,K. Wang, Xu, Yu, Zheng, Zhong, etc. considered Zhu’s conjecture,they found that Zhu’s conjecture does not hold in general, and it ismodified as follows: Mφ has at most n distinct minimal reducingsubspaces on L2a.

In 2012, Douglas, Putinar and K. Wang proved the modifiedconjecture.


Let Aφ = {Mφ,M∗

φ}′ = {A ∈ B(L2a) : AMφ = MφA,AM

∗

φ = M∗

φA}.Then Aφ is a von Neumann algebra.


Let Aφ = {Mφ,M∗


∗

φ = M∗

φA}.Then Aφ is a von Neumann algebra.Douglas, Putinar and K. Wang showed that the von Neumann algebraAφ is commutative of dimension q, where q is the number ofconnected components of the Riemann surface Sφ for φ−1 ◦ φ.


Let Aφ = {Mφ,M∗


∗

φ = M∗


On D: when n = 2, under the norm ‖ · ‖1, L. K. Zhao (2009) showedthat Mφ is reducible if and only if φ is equivalent to z2, i.e.φ = ϕλ(z

2), λ ∈ D, where ‖f ‖21 = |f (0)|2 +∫D|f ′|2dA.


Let Aφ = {Mφ,M∗


∗

φ = M∗



2), λ ∈ D, where ‖f ‖21 = |f (0)|2 +∫D|f ′|2dA.

Under the norm ‖ · ‖D , when n = 2, Chen and Lee (2014) also provedthat Mφ is reducible if and only if φ is equivalent to z2, where‖f ‖2D = ‖f ‖2

H2 +∫D|f ′|2dA.


Let Aφ = {Mφ,M∗


∗

φ = M∗



2), λ ∈ D, where ‖f ‖21 = |f (0)|2 +∫D|f ′|2dA.

Under the norm ‖ · ‖D , when n = 2, Chen and Lee (2014) also provedthat Mφ is reducible if and only if φ is equivalent to z2, where‖f ‖2D = ‖f ‖2

H2 +∫D|f ′|2dA.

When n ≥ 3, it is unknown when Mφ is reducible on either Dirichletspace with the norm ‖ · ‖1 or ‖ · ‖D .


Main Results

Let U : D → L2a, Uf = (zf )′, then U is a unitary operator.

Theorem 2.1 (L)

Let φ be a finite Blaschke product. If M is a reducing subspace of Mφ on

D, then UM = (zM)′ is a reducing subspace of Mφ on L2a.


Main Results

Let U : D → L2a, Uf = (zf )′, then U is a unitary operator.

Theorem 2.1 (L)

Let φ be a finite Blaschke product. If M is a reducing subspace of Mφ on

D, then UM = (zM)′ is a reducing subspace of Mφ on L2a.

We say that two Blaschke products φ1 and φ2 are equivalent if there existλ ∈ D such that φ2 = ϕλ ◦ φ1.

Theorem 2.2 (L)

Let φ be a finite Blaschke product of order 3. Then Mφ is reducible on D

if and only if φ is equivalent to z3.


Theorem 2.3 (Gu-Xiao-L)

Let φ be a finite Blaschke product of order n = 5 or 7. Then Mφ is

reducible on D if and only if φ is equivalent to zn.


Let E = {β ∈ D : ∃ α ∈ D, φ′(α) = 0 and φ(α) = φ(β)}, then E is a finiteset, and φ−1 ◦ φ is an n-branched analytic function defined and arbitrarilycontinuable in D \ E .


Let E = {β ∈ D : ∃ α ∈ D, φ′(α) = 0 and φ(α) = φ(β)}, then E is a finiteset, and φ−1 ◦ φ is an n-branched analytic function defined and arbitrarilycontinuable in D \ E .For an open set V ⊆ D, a local inverse of φ in V is a function ρ analyticin V which satisfies ρ(V ) ⊆ D and φ(ρ(z)) = φ(z) on V . Therefore thereare n local inverses ρ0, ρ1, · · · , ρn−1 for φ in D \ E , i.e.

φ−1 ◦ φ = {ρ0(z), ρ1(z), · · · , ρn−1(z)}.


Let E = {β ∈ D : ∃ α ∈ D, φ′(α) = 0 and φ(α) = φ(β)}, then E is a finiteset, and φ−1 ◦ φ is an n-branched analytic function defined and arbitrarilycontinuable in D \ E .For an open set V ⊆ D, a local inverse of φ in V is a function ρ analyticin V which satisfies ρ(V ) ⊆ D and φ(ρ(z)) = φ(z) on V . Therefore thereare n local inverses ρ0, ρ1, · · · , ρn−1 for φ in D \ E , i.e.

φ−1 ◦ φ = {ρ0(z), ρ1(z), · · · , ρn−1(z)}.

We say that ρi ∼ ρj if there is a loop γ in D\E such that ρi and ρj areanalytic continuation of each other along γ. Then ∼ is an equivalencerelation. Using this equivalence relation, we obtain a partition{G1,G2, · · · ,Gq} for {ρ0, ρ1, · · · , ρn−1}.


Defineξi f (z) =

∑

ρ∈Gi

f (ρ(z))ρ′(z) z ∈ D \ E , f ∈ L2a.


Defineξi f (z) =

∑

ρ∈Gi

f (ρ(z))ρ′(z) z ∈ D \ E , f ∈ L2a.

Theorem 2.4 (Douglas-S. H. Sun-Zheng, 2011)

Let φ be a finite Blaschke product. The von Neumann algebra Aφ is

generated by the linearly independent operators ξ1, · · · , ξq and hence has

dimension q.


Defineξi f (z) =

∑

ρ∈Gi

f (ρ(z))ρ′(z) z ∈ D \ E , f ∈ L2a.

Theorem 2.4 (Douglas-S. H. Sun-Zheng, 2011)

Let φ be a finite Blaschke product. The von Neumann algebra Aφ is

generated by the linearly independent operators ξ1, · · · , ξq and hence has

dimension q.

Theorem 2.5 (Douglas-Putinar-K. Wang, 2012)

Let φ be a finite Blaschke product. Then The von Neumann algebra Aφ is

commutative of dimension q.


Let Aφ = {Mφ,M∗

φ}′ ⊂ B(D). Recall that U : D → L2a, Uf = (zf )′ is a

unitary operator.If M is a reducing subspace of Mφ on D, then UM = (zM)′ is areducing subspace of Mφ on L2a.

Lemma

Let T ∈ Aφ, f ∈ D. Then there are a1, · · · , aq ∈ C such that

Tf (z) =

q∑

i=1

aiFi(z)− Fi (0)

z, T ∗f (z) =

q∑

i=1

aiHi (z)− Hi (0)

z

where Fi(z) =∑ρ∈Gi

f (ρ(z))ρ(z), Hi (z) =∑

ρ∈G−1i

f (ρ(z))ρ(z) and

G−1i = {ρ : ρ−1 ∈ Gi}.


Let

L = span

{(a1, · · · , aq) : f ∈ D,

q∑

i=1

aiFi (0) = 0,

q∑

i=1

aiHi (0) = 0

}.


Let

L = span

{(a1, · · · , aq) : f ∈ D,

q∑

i=1

aiFi (0) = 0,

q∑

i=1

aiHi (0) = 0

}.

Theorem 2.7 (Gu-Xiao-L)

Aφ is a commutative von Neumann algebra, and dim Aφ = dimL.


Note that the family of local inverses {ρ0, · · · , ρn−1} has a group-likeproperty under composition near the boundary of D. Write j ∈ Gk ifρj ∈ Gk , then {G1,G2, · · · ,Gq} is a partition of the additive groupZn = {0, 1, · · · , n − 1}.


Note that the family of local inverses {ρ0, · · · , ρn−1} has a group-likeproperty under composition near the boundary of D. Write j ∈ Gk ifρj ∈ Gk , then {G1,G2, · · · ,Gq} is a partition of the additive groupZn = {0, 1, · · · , n − 1}.Necessary conditions for the partitions {G1,G2, · · · ,Gq}.

(A1) One of {Gk} is {0} since ρ0(z) = z .

(A2) For each Gj = {j1, · · · , jm}, there exists k such that

Gk = G−1j = {n − j1, · · · , n − jm}.

(A3) For any Gj ,Gk , there are Gl1 , · · · ,Glm such that

Gj + Gk = Gl1 ∪ · · · ∪ Glm counting multiplicities on both sides.


Let φ be a finite Blaschke product of order 5. We have the followingpossible partitions {G1,G2, · · · ,Gq}. By Corollary 8.4 of Douglas-Putinarand K. Wang, q 6= 4. We have the following cases.



(i) If q = 5, then the partition is {{0}, {1}, {2}, {3}, {4}}.




(ii) If q = 3, without loss of generality, suppose G1 = {0}. Letm = min{#G2,#G3}. By condition (A3), m can not be 1. Thus m = 2,then #G2 = #G3 = 2, and there are essentially three cases.

(a) G2 = {1, 2},G3 = {3, 4};

(b) G2 = {1, 3},G3 = {2, 4};

(c) G2 = {1, 4},G3 = {2, 3}.

Case (a) doesn’t satisfy condition (A3), since G2 + G2 = {2, 3, 3, 4}.Similarly, case (b) doesn’t satisfy condition (A3). So we have the possiblepartition (c).




(ii) If q = 3, without loss of generality, suppose G1 = {0}. Letm = min{#G2,#G3}. By condition (A3), m can not be 1. Thus m = 2,then #G2 = #G3 = 2, and there are essentially three cases.

(a) G2 = {1, 2},G3 = {3, 4};

(b) G2 = {1, 3},G3 = {2, 4};

(c) G2 = {1, 4},G3 = {2, 3}.

Case (a) doesn’t satisfy condition (A3), since G2 + G2 = {2, 3, 3, 4}.Similarly, case (b) doesn’t satisfy condition (A3). So we have the possiblepartition (c).

(iii) If q = 2, then the partition is {{0}, {1, 2, 3, 4}}.


Therefore when n = 5, the possible partitions are

{{{0}, {1}, {2}, {3}, {4}}, {{0}, {1, 4}, {2, 3}};

{{0}, {1, 2, 3, 4}}.


Therefore when n = 5, the possible partitions are

{{{0}, {1}, {2}, {3}, {4}}, {{0}, {1, 4}, {2, 3}};

{{0}, {1, 2, 3, 4}}.

Theorem (Gu-Xiao-L)

Let φ be a finite Blaschke product of order 5. Then one of the following

holds:

(a) If φ is equivalent to ϕ5α, α ∈ D, then the partition is

{{0}, {1}, {2}, {3}, {4}};

(b) If φ is equivalent to

(z2ϕ2αϕβ) ◦ ϕγ , α, β ∈ D\{0}, γ ∈ D, α/β ∈ R, ϕβ(α) =

α2

β, then the

partition is {{0}, {1, 4}, {2, 3}};

(c) If φ is not equivalent to any of the functions in (a) and (b), then the

partition is {{0}, {1, 2, 3, 4}}.


Thank You!


Shuaibing Luo, Hunan University Joint work with Caixing Gu ... · ReducingsubspacesoftheDirichletspace Shuaibing Luo, Hunan University Joint work with Caixing Gu (California Polytechnic

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