the Commutant of some Multiplication Operators on the Sobolev Disk Algebra

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Volume 2012, Article ID 378217,17pages doi:10.1155/2012/378217

Research Article

A Similarity Invariant and

the Commutant of some Multiplication Operators on the Sobolev Disk Algebra

Ruifang Zhao

Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China

Correspondence should be addressed to Ruifang Zhao,[email protected] Received 31 March 2012; Accepted 19 July 2012

Academic Editor: Gideon Schechtman

Copyrightq2012 Ruifang Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

LetRDbe the algebra generated in Sobolev spaceW²²Dby the rational functions with poles outside the unit diskD. In this paper, we study the similarity invariant of the multiplication operatorsMg inLRD, whengis univalent analytic onDorMgis strongly irreducible. And the commutants of multiplication operators whose symbols are composite functions, univalent analytic functions, or entire functions are studied.

1. Introduction

LetΩbe an analytic Cauchy domain in the complex plane and letW²²Ωdenote the Sobolev space,

W²²Ω

f ∈L²Ω, dA: the distributional partial derivatives of first and second order of f belong toL²Ω, dA

, 1.1

dAdenotes the planar Lebesgue measure. Forf, g∈W²²Ω, we define

f, g

|α|≤2

D^αfD^αgdA. 1.2

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ThenW²²Ωis a Hilbert space and a Banach algebra with identity under an equivalent norm.

W²²Ωcan be continuously embedded in the spaceCΩof continuous functions onΩby Sobolev embedding theorem.

Let RΩ be the subalgebra generated by the rational functions with poles outside Ω. When Ω D, the unit disc, we call RD Sobolev disk algebra. For f ∈ RD, the multiplication operator Mf on RD is defined byMfg fg,g ∈ RD. Then RD AMze0, wheree0is the identity inRDandAMzis the algebra generated byMzand identity. In fact, RD consists of all analytic functions in W²²D. We have the following properties of the spaceRDand the multiplication operators on it.

Proposition 1.1see1. (i) Hilbert spaceRDhas an orthogonal basis{en}^∞_n0, where

enβnzⁿ, βn

n1 3n⁴−n²2n1

π _1/2

, n0,1,2, . . . . 1.3

(ii) As a functional Hilbert space,RDhas reproducing kernel which is

ku, v ^∞

n0

β²_nuⁿvⁿ. 1.4

Then forz₀∈D,

kz0 ^∞

n0

β²_nzⁿ₀

zⁿ. 1.5

(iii) Iffz _∞

n0f_nzⁿis analytic onD, thenfz ∈RDif and only if_∞

n0|fn|²/β²_n <

∞.

(iv) The operatorMf admits the following matrix representation with respect to{en}^∞_n0:

M_f

⎛

⎜⎜

⎜⎝ c₀ c1

β0

β₁ c0 0 c₂β₀

β₂ c₁β₁ β₂ c₀ c3

β0

β3 c2

β1

β3 c1

β2

β3 c0

... ... . .. ... ...

⎞

⎟⎟

⎟⎠

. 1.6

Note thatRDis a subset of the disk algebraAD, hence a subset ofH^∞. Because of the special definition of the inner product and the complex behavior of the boundary value, the structure of the spaceRDis much more complicated thanH^∞orH². For more details about the Sobolev disk algebra, the reader refers to1–3.

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Let H be a complex separable Hilbert space and LH denote the collection of bounded linear operators onH. One of the basic problems in operator theory is to determine when two operators A and B in LH are similar. A quantity quantities or a property properties P is similarity invariant invariantsif AhasP andA ∼ Bimplies thatBhas P 2. From this point of view, one of the basic problems in operator theory mentioned above is to determine the similarity invariants. There have already been a lot of results on the similarity invariants of operators, especially that of nonadjoint operators, which can be found in, for example,4–6. In7, Wang et al. proved that inRD,Mf is similar toMzⁿ if and only iff is ann-Blaschke product. In this paper, we study the similarity invariant of the multiplication operatorsMginLRD, whengis univalent analytic onDorMgis strongly irreducible.

It is well known that the commutant of a bounded linear operator or operators on a complex, separable Hilbert space plays an important role in determining the structure of this operator or these operators. The commutant of a multiplication operator on Sobolev disk algebra has been studied in the literature see 1–3. In this paper, we describe the commutant of the multiplication operatorM_fg whengis ann-Blaschke product. And by this result, we generalize the result which is obtained by Liu and Wang in3. Moreover, we study the commutants of the multiplication operators whose symbols are composite functions, univalent analytic functions, or entire functions.

2. Similarity Invariant of Some Multiplication Operators

In this section, we will characterize the similarity invariant of some multiplication operators on Sobolev disk algebra. Here, we briefly recall some background information.

For T in LH, let σT, σ_pT, andσ_eT be the spectrum, point spectrum, and essential spectrum of a bounded linearly operatorT, respectively. An operatorAinLH is said to be a Cowen-Douglas operator with index n if there exists Ω, a connected open subset of complex planeC, andn, a positive integer, such that

i Ω⊂σA {λ∈C:A−λis not invertible};

iiranA−λ:{y;A−λxy, x∈ H}HforλinΩ;

iiinulA−λ:dim kerA−λ nforλinΩ;

iv

{kerA−λ:λ∈Ω}H,

whereivis equivalent toiv8;

ivthere existsλ₀inΩ, such that

{kerA−λ₀^k:k≥1}H.

BnΩdenotes the collection of Cowen-Douglas operators with indexn.

ForT ∈ LH, the set of operators which commute with it isAT. That isAT {A∈ LH:AT TA}. OperatorTis strongly irreducible if there is no nontrivial idempotent inAT 8,9. DenoteSIthe set of all strongly irreducible operators onH.

Definition 2.1. LetHbe a Hilbert space andA, Bbe inLH.τ_AB is said to be a Rosenblum operator onLHif for arbitraryC∈ LH, τABC AC−CB.

Lemma 2.2see1. Letfbe inRD, then iσMf fD;

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iiσeMf f∂D, where∂Ddenotes the boundary of the unit discD;

iiiletz₀ ∈ Dandfz0 /∈ f∂D. Denote the component ofρ_s−FMfcontainingfz0as Ω, thenM^∗_f ∈ BnΩ, wherenis the number of the zeros offz−fz0inD.

Lemma 2.3see10. Setf ∈RDandBaz z−a/1−az, a∈D. ThenM_f◦B_a ∼Mf. Theorem 2.4. Letf andgbe inRDand be univalent and analytic onD. ThenM_f ∼M_g if and only iffD gD.

Proof. “⇒”: SetMf ∼Mg. ByLemma 2.2, we have

f D

σ Mf

σ Mg

g D

, fT σe Mf

σe Mg

gT, 2.1

whereTis the unit circle. Sincefandgare univalent and analytic onD, then fT ∂f

D ∂g

D

gT. 2.2

Therefore,

fD f D

\fT g D

\gT gD. 2.3

“⇐”: SetfD gD Ω. Becausegis univalent analytic fromDtoΩ,g⁻¹ :Ω → D is also univalent analytic. Theng⁻¹◦f is injective and surjective analytic function onD. If g⁻¹◦fz0 0, there exists a M ¨obius transformϕwithϕz0 z−z0/1−z0zand a complex numbercwith|c|1 such thatg⁻¹◦fz cϕ_z₀see11. Thereforefz gcϕz0z. By Lemma 2.3,M_f ∼M_g.

Lemma 2.5see3. Givenf ∈RD, the following are equivalent:

iM^∗_f ∈ B1Ω;

iiAMf {Mg:g∈RD};

iiiM_f ∈SI.

Theorem 2.6. Letf, g∈RDandfis univalent analytic onD.Mf ∼Mgif and only if there exists a functionχcz−z0/1−z0zsuch thatfg◦χ, wherez0∈Dandc∈C, |c|1.

Proof. “⇒”: Suppose thatg is not univalent onD. There exists someλ ∈ gDsuch that the number of zeros ofgz−λonDisn >1. ByLemma 2.2,M^∗_g∈ BnΩwhereΩis a connected open subset ofgD. SinceMf ∼Mg, we haveM^∗_f ∈ BnΩ. This contradicts toMf that is a strongly irreducible operatorseeLemma 2.5. Sog is univalent analytic onD. By the proof ofTheorem 2.4, there exists a functionχ cz−z0/1−z0zsuch thatf g◦χ, where z0∈Dandc∈C,|c|1.

“⇐”: By the conditions of the theorem, g is univalent analytic on D. Since fD g◦χD gD, we haveMf ∼MgbyTheorem 2.4.

For any operatorT on Hilbert spaceHand any integern, 1 ≤n≤ ∞, letTⁿ denote the direct sum ofncopies ofTonHⁿH ⊕ · · · ⊕ H.

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Lemma 2.7see2. LetA1, A2 be strongly irreducible Cowen-Douglas operators. Assume that A1/∼A2andT Aⁿ₁¹⊕Aⁿ₂², wheren1, n2 are natural numbers. Then for each maximal idealJof AT,Jmust be one of the following two forms:

i J

⎛

⎝ J11 kerτ_Aⁿ1 1 , Aⁿ₂²

kerτ_An2

2 , Aⁿ₁¹ A Aⁿ₂²

⎞

⎠;

ii J

⎛

⎝ A Aⁿ₁¹

kerτ_An1 1 , Aⁿ₂²

kerτ_An2

2 , Aⁿ₁¹ J22

⎞

⎠,

2.4

whereJiiis a maximal ideal ofAAⁿ_i ⁱ,i1,2.

Theorem 2.8. Letf, g∈RDandMf, Mg∈SI. The following statements are equivalent:

iM_f ∼M_g;

iithere existX₁, X₂, . . . , X_n∈kerτ_M_f_{, M}_g andY₁, Y₂, . . . , Y_n∈kerτ_M_g_{, M}_f such that

X₁Y₁X₂Y₂· · ·X_nY_nid_RD, Y₁X₁Y₂X₂· · ·Y_nX_nid_RD, 2.5

whereid_RDdenotes the identity operator ofRD.

Proof. i⇒ ii : LetMf ∼Mg. SetXbe invertible inLRDandMfX XMg. ThenX andX⁻¹are what we want.

ii ⇒ i : Since M_f and M_g are in SI, we have M_f^∗ and M^∗_g that are strongly irreducible and inB1ΩbyLemma 2.5. Computations show

A

M_f^∗⊕M^∗_g

⎛

⎝ A M^∗_f

kerτM^∗_f, M^∗g

kerτ_M^∗_g_{, M}^∗

f A

M^∗_g

⎞

⎠. 2.6

Suppose thatM_f^∗/∼M_g^∗. ByLemma 2.7, each maximal idealJofAM_f^∗⊕M^∗_gmust be one of the following two forms

J

J11 kerτM^∗_f, M^∗g

kerτM^∗g, M^∗_f A

M^∗_g

or J

A M^∗_f

kerτ_M^∗

f, Mg^∗

kerτ_M^∗_g_{, M}^∗

f J22

, 2.7

where J11 and J22 are the maximal ideals of AM^∗_f and AM^∗_g, respectively. We can assume thatJadmits the first form. Then

0 Y_i^∗ X^∗_i 0

∈ J, i1,2, . . . , n 0 Y_i^∗

X^∗_i 0

0 Y_i^∗ X^∗_i 0

Y_i^∗X_i^∗ 0 0 X^∗_iY_i^∗

∈ J.

2.8

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It follows that

n i1

Y_i^∗X^∗_i 0 0 X^∗_iY_i^∗

id_RD 0 0 id_RD

∈ J. 2.9

This contradicts thatJis a maximal ideal. SoM^∗_f ∼M^∗_gandMf ∼Mg.

3. The Commutant Algebra of Multiplication Operator

In3, Liu and Wang give the following proposition.

Proposition 3.1. Letfz zⁿhz∈RD,hz/0,z∈D. ThenAMf AMzⁿ∩AMh. LetBnz _n

k1z−ak/1−akz,|ak| < 1, k 1,2, . . . , nben-Blaschke product.

Consideringzⁿis a specialn-Blaschke product, we study the commutant ofM_f wherefz B_nzhz. The following theorem is obtained, and by this result, the above proposition is generalized.

Theorem 3.2. Letf Bnh ∈ RDwith h∈ RDwhere hdoes not vanish onD. If there exists 0/λ ∈ Csuch thath−λcan be divided by eachz−ak/1−akz, thenAMf AMBn∩ AMh.

To prove the above theorem, we need the following lemmas.

Lemma 3.3see7. Giveng ∈ RD, Mg ∼ M_zⁿ∼Mzⁿif and only ifg is ann-Blaschke product.

Lemma 3.4see12. LetN be a nilpotent operator onH and let X0 λN, 0/λ ∈ C. If B, A₀, A₁, . . .∈BHsatisfyAk ≤Mk0,1,2, . . .andA_kX₀X₀A_k−1B, k1,2,3. . ., thenA₀A₁A₂· · ·.

Now we will proveTheorem 3.2.

Proof. From h ∈ RD, we have M_f^∗ M^∗_B

nM^∗_h. Then, from Lemma 3.3, there exists an invertible operatorX∈ LRD,⊕ⁿ₁RDsuch thatXM^∗_B

nX⁻¹⊕ⁿ₁M^∗_z. It follows that AXM_B^∗_nX⁻¹XM^∗_hX⁻¹ ⊕ⁿ₁M_z^∗

T, 3.1

whereT XM^∗_hX⁻¹∈ A⊕ⁿ₁M^∗_z. So we only need to prove AA A ⊕ⁿ₁M^∗_z

∩ AT. 3.2

Since

⊕ⁿ₁RD ker ⊕ⁿ₁M^∗_z

⊕

ker ⊕ⁿ₁M_z^∗2−ker ⊕ⁿ₁M^∗_z

⊕

ker ⊕ⁿ₁M^∗_z3−ker ⊕ⁿ₁M^∗_z2

⊕ · · · ⊕

ker ⊕ⁿ₁M^∗_zk−ker ⊕ⁿ₁M^∗_z_k−1

⊕ · · · 3.3

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we have

⊕ⁿ₁M^∗_z

⎛

⎜⎜

⎝ 0 β0

β1

I_n 0 0 · · ·

0 0 β1

β₂In 0 · · ·

0 0 0 β₂

β₃I_n · · · ... ... ... ... . ..

⎞

⎟⎟

⎠

, 3.4

whereI_nis ann×nidentity matrix. Computations show thatG∈ A⊕ⁿ₁M^∗_zif and only if

G

⎛

⎜⎜

⎝ G1

β0

β₁G12

β0

β₂G13

β0

β₃G14 · · · β0

β_k−1G1,k · · · 0 G₁ β₁

β₂G₁₂ β₁

β₃G₁₃ · · · β₁

β_k−1G_1,k−1 · · ·

0 0 G1

β2

β3G12 · · · β2

β_k−1G_1,k−2 · · · ... ... ... ... . .. ... · · ·

0 0 0 0 · · · β_k−2

β_k−1G_1,2 · · · ... ... ... ... · · · ... . ..

⎞

⎟⎟

⎠

. 3.5

Therefore,

T

⎛

⎜⎜

⎝ T1

β₀ β₁T12

β₀ β₂T13

β₀

β₃T14 · · · β₀

β_k−1T1,k · · · 0 T₁ β₁

β₂T₁₂ β₁

β₃T₁₃ · · · β₁

β_k−1T_1,k−1 · · ·

0 0 T1

β2

β3T12 · · · β2

β_k−1T_1,k−2 · · · ... ... ... ... . .. ... · · ·

0 0 0 0 · · · β_k−2

β_k−1T1,2 · · · ... ... ... ... · · · ... . ..

⎞

⎟⎟

⎠

. 3.6

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Setgk z−ak/1−akz, 1 ≤ k ≤ n, andBn g1g2· · ·gn. Then for eachgk, there existsϕ_k∈RDsuch thath−λg_kϕ_k,1≤k≤n. So

T XM^∗_hX⁻¹XM^∗_g_kX⁻¹XM^∗_ϕ_kX⁻¹λid_⊕ⁿ₁_RDVkSkλid_⊕ⁿ₁_RD, 3.7 whereV_k XM_g^∗_kX⁻¹, S_k XM_ϕ^∗_kX⁻¹. Fork 1,2,3, . . . , n, j 1,2,3, . . . , n, M_ϕ_k, andM_g_j pairwise commute. Hence,Vk, Sjpairwise commute too. SinceT1T|ker⊕ⁿ₁M^∗_zand

ker ⊕ⁿ₁M_z^∗

kerXM^∗_B

nX⁻¹kerV₁V₂· · ·V_n 3.8 for allx∈ker⊕ⁿ₁M^∗_z,

T−λid_⊕ⁿ

1RD

_n

xV₁S₁V₂S₂· · ·V_nS_nxS₁S₂· · ·S_nV₁V₂· · ·V_nx0. 3.9

Therefore,T1−λInis nilpotent operator. Now we setNT1−λIn, that isT1λInN. So

A⊕ⁿ₁M^∗_zT

⎛

⎜⎜

⎝ 0 β0

β1

λI_nN β0

β2

T₁₂ β0

β3

T₁₃ · · · β0

βk

T_1,k· · ·

0 0 β₁

β₂

λInN β₁

β₃T12 · · · β₁ β_kT_1,k−1

0 0 0 β₂

β3

λI_nN

· · · β₂ βk

T_1,k−2 · · ·

... ... ... ... . .. ... · · ·

0 0 0 0 · · · β_k−1

β_k

λI_nN

· · ·

... ... ... ... · · · ... . ..

⎞

⎟⎟

⎠

. 3.10

So we only need to prove thatAA⊂ A⊕ⁿ₁M^∗_z. In fact, ifQ∈ AA⊂ A⊕ⁿ₁M^∗_z,

QAAQ⇒QT ⊕ⁿ₁M^∗_z

T ⊕ⁿ₁M^∗_z Q ⇒QT ⊕ⁿ₁M^∗_z

TQ ⊕ⁿ₁M^∗_z ⇒QT−TQ ⊕ⁿ₁M_z^∗

0 ⇒ ⊕ⁿ₁Mz

T^∗Q^∗−Q^∗T^∗ 0.

3.11

It follows from ker⊕ⁿ₁M_z {0}thatT^∗Q^∗Q^∗T^∗. Namely,TQQT.

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We are now in need to prove thatAA⊂ A⊕ⁿ₁M^∗_z. Suppose thatQ∈ AA. Since hdoes not vanish onD, ranMhRDand so kerT{0}. BecauseA ⊕ⁿ₁M^∗_zT T⊕ⁿ₁M^∗_z,

kerAker ⊕ⁿ₁M_z^∗

, kerA^kker ⊕ⁿ₁M^∗_zk

. 3.12

It follows fromQAAQthat kerAand kerA^kare both the invariant subspaces ofQ. Since

⊕ⁿ₁RD kerA⊕

kerA²kerA

⊕

kerA³kerA²

⊕ · · · ⊕

kerA^kkerA^k−1

⊕ · · ·, 3.13 Aadmits the matrix representation3.5with the above decomposition. So

Q

⎛

⎜⎜

⎜⎝

Q₁ Q₁₂ Q₁₃ Q₁₄ · · · 0 Q2 Q23 Q24 · · · 0 0 Q₃ Q₃₄ · · · ... ... ... ... ...

⎞

⎟⎟

⎟⎠. 3.14

FromQAAQ, we have

⎛

⎜⎜

⎝ 0 β0

β1

Q₁

λI_nN β0

β2

Q₁T₁₂β1

β2

Q₁₂

λI_nN

· · ·

0 0 β1

β₂Q2

λInN

· · ·

... ... ... ...

⎞

⎟⎟

⎠

⎛

⎜⎜

⎝ 0 β₀

β₁

λInN Q2

β₀ β₁

λInN

Q23β₀

β₂T12Q3 · · ·

0 0 β₁

β2

λI_nN

Q₃ · · ·

... ... ... ...

⎞

⎟⎟

⎠ .

3.15

Comparing then, n1entries of both sides, we haveQ₁ Q₂ Q₃ · · · byLemma 3.4.

Comparing then, n2entries ofQAandAQ, we have β_n1

β_n

λI_nN

Q_n1,n2 β_n

β_n−1Q_n,n1

λI_nN

Q₁T₁₂−T₁₂Q₁. 3.16

It follows fromLemma 3.4that β_n1

β_n Q_n1,n2 β_n

β_n−1Q_n,n1. 3.17

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SettingQ₁₂ β1/β0Q12, we have

Q_n,n1 β1β_n−1

β0βn Q12 β_n−1

βn Q₁₂ . 3.18

Inductively, if Q_n,nk−1 βn−1/β_nk−2Q_1,k, where Q_1,k βk−1/β₀Q1k, we need to prove Q_n,nk βn−1/β_nk−1Q_1,k1. Comparing then, nk1entries ofQAandAQ, we have

β_nk βn

λI_nN

Q_n1,nk1 β_nk−1

β_n−1 Q_n,nk

λI_nN

T₁₂Q_1,k−Q_1,kT₁₂

T13Q_1,k−1−Q_1,k−1T13

· · · T_1,k1Q1−Q1T_1,k1.

3.19

Therefore, byLemma 3.4,

β_nk

β_n Q_n1,nk1 β_nk−1

β_n−1 Q_n,nk. 3.20

Computations showQ_n,nk β_n−1/β_nk−1Q_1,k1. SinceQis the form of3.5,Q∈ A⊕ⁿ₁M^∗_z. SoAMf AMBn∩ AMh.

Corollary 3.5. Letf B_nh ∈RDwithh ∈ RDwherehdoes not vanish onD. If there exists 0/λ∈Cthath−λcan be divided byB_n, thenAMf AMBn∩ AMh.

By the following lemma, we discuss the commutant of the multiplication operators whose symbols are composite functions inRD.

Lemma 3.6. ForTinLRDandfinRDthe following are equivalent:

iT ∈ AMf;

iifor allα∈D,T^∗k_α⊥f−fαRD;

iiithere is a setJ⊂Dsuch that

α∈J1− |α| ∞and for allα∈J,T^∗kα⊥f−fαRD.

Proof. i⇒iiLetT ∈ AMf. For allg∈RDandα∈D,we have

f−fα g, T^∗kα

TMfg, kα>−fα< Tg, kα

M_fTg, k_α

−fα Tg, k_α fα Tg

α−fα Tg α 0.

3.21

ii⇒iiiLetJ D.

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iii⇒iLetT ∈ LRDwithT^∗kα⊥f−fαRDfor allα∈J. Forg∈RDand α∈J, we have

0 f−fα g, T^∗kα

TMfg

α−fα Tg α TM_fg

α− M_fTg α.

3.22

SinceJis not a Blaschke sequence, this meansTMfg MfTg. ThereforeTMf MfT.

Lemma 3.7see13. Supposef :G → Ωis a surjective analytic function and for eachξ ∈ Ω, nξis the number of points inf⁻¹ξ. Then

G

f²dA

ΩnξdAξ. 3.23

Proposition 3.8. Letf be inRDandf is analytic onD. Suppose for eachξ ∈ fD, there aren points inf⁻¹ξ. Then forg∈RfD, we haveAMf⊂ AM_g◦f.

Proof. By the Embedding Theory of Sobolev space,g◦f∈CD. Therefore

D

g fz²dAz≤πg

∞. 3.24

ByLemma 3.7,

D

g fz ²dAz

D

g fz²fz²dAz n

fD

gω²dAω

≤ng

RfD.

3.25

Sincefis analytic onD,fz,fzis bounded onD. Hence

D

g fz²fz⁴dAz≤fz

∞

D

g fz²fz²dAz

≤nfz

∞g

RfD,

D

g fz²fz²dAz≤fz

∞g

RfD.

3.26

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Therefore,

D

g fz ²dAz<∞. 3.27

By3.24,3.25, and3.27, we haveg◦f∈RD.

For allβ∈fD, becausegz−gβ/z−β∈RfD,g◦fz−β/fz−β∈ RD. For allα∈D

g fz

−g fα

g fz

−g fα

fz−fα fz−fα

, 3.28

so that

g fz

−g fα

RD⊂ fz−fα

RD. 3.29

SetT ∈ AMf. ByLemma 3.6, for allα∈D,T^∗k_α ⊥ f−fαRD. Hence,T^∗k_α ⊥ gfz−gfαRDand we haveT ∈ AMg◦f.

Corollary 3.9. IfB_nz _n

i1z−ai/1−aiz,ai/a_j, i /j, |ai|<1andf∈RD,AMBn⊂ AMf◦Bn.

Proposition 3.10. Letf be in RDand Bn is an n-Blaschke product. If AMf AMz

{Mg;g∈RD}, thenAM_f◦B_n AMBn.

Proof. From 3, we know that AMz {Mg;g ∈ RD}. ByLemma 3.3, MBn ∼ ⊕ⁿ₁Mz. Then there exists an invertible operatorXinLRD,⊕ⁿ₁RDsuch thatXMBnX⁻¹ ⊕ⁿ₁Mz. SinceM_f◦B_n fMBn, we haveXM_f_◦B_nX⁻¹ ⊕ⁿ₁M_f. Therefore, we will only prove that A⊕ⁿ₁M_f A⊕ⁿ₁M_z. SetT∈ A⊕ⁿ₁M_fand

T

⎛

⎜⎝

T₁₁ T₁₂ · · · T_1n T12 T13 · · · T2n

... ... · · · ...

⎞

⎟⎠ RD RD

...

. 3.30

Since

T ⊕ⁿ₁M_f

⎛

⎜⎝

T11 T12 · · · T1n

T₁₂ T₁₃ · · · T_2n ... ... · · · ...

⎞

⎟⎠

⎛

⎜⎝

Mf 0 · · · 0 M_f · · ·

0 . ..

⎞

⎟⎠

⊕ⁿ₁M_f T

⎛

⎜⎝

Mf 0 · · · 0 M_f · · ·

0 . ..

⎞

⎟⎠

⎛

⎜⎝

T11 T12 · · · T1n

T₁₂ T₁₃ · · · T_2n ... ... · · · ...

⎞

⎟⎠,

3.31

we have TijMf MfTij for i, j 1,2, . . .. SoTij ∈ AMf AMz and A⊕ⁿ₁Mf ⊂ A⊕ⁿ₁M_z. Similarly,A⊕ⁿ₁M_z A⊕ⁿ₁M_f. HenceAMf◦Bn AMBn.

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Letfbe an injective function inRDandΩ fD. Then for eachz0∈D, it is obvious thatfz0is not infT. Denote the component ofρ_s−FMfcontainingfz0asΩ, thenz₀ is the only zero point offz−fz0inD. ByLemma 2.2,M_f^∗ is a Cowen-Douglas operator with index 1. ByLemma 2.5, we haveAMf {Mf;f∈RD}AMz. So the following corollary is obtained.

Corollary 3.11. Letf be a univalent analytic function inRDand B_n be ann-Blaschke product.

ThenAMBn AM_f◦Bn.

Lemma 3.6shows iffis inRDandz0is inD, then

!"

T^∗kz0 :T ∈ A Mf

#⊂ f−fz0

RD ^⊥kerM^∗_f−f_z₀. 3.32

Easy examples show that kerM^∗_f−f_z

0 ⊂

{T^∗k_z₀ : T ∈ AMf}is not true. The following proposition shows that iffz zⁿ,

{T^∗k_z₀:T ∈ AMf}kerM^∗_f−fz

0. Lemma 3.12see1. Letf ∈RD, M^∗_f ∈ BnΩ, z0 ∈D1:f⁻¹Ω, and

fz−fz0 z−z₀^h¹z−z₁^h²· · ·z−z_l^h^l1g_z₀z, 3.33 where{zi}^l_i1 ⊂D₁ are pairwise distinct,_l1

i1h_i n, g_z₀z/0, z ∈D. Choosek¹_z_i, . . . , k_z^h_iⁱ¹⁻¹ ∈ RDsuch that

M^∗_z−z_ik_z¹_i k_z_i, . . . , M^∗_z−z_ik_z^h_iⁱ¹⁻¹k_z^h_iⁱ¹⁻², 0≤i≤l. 3.34

Then there exists a linearly independent set

K_z₀:$

k_z₀, k¹_z₀, . . . , k^h_z¹₀⁻¹, k_z₁, . . . , k^h_z₁²⁻¹, . . . , k^h_z_l^l1⁻¹%

, 3.35

such that kerM^∗_f−fz

0

Kz0.

Letn ≥ 2 andω be thenth root of 1, that is,ω ∈ Candωⁿ 1. LetΔn denote the Vandermonde determinant of ordern:

Δn

1 1 1 · · · 1

1 ω ω² · · · ωⁿ⁻¹ ... ... ... ... 1 ωⁿ⁻¹ ω²ⁿ⁻¹ · · · ωⁿ⁻¹²

. 3.36

For 1≤i,j≤n, thei, j-cofactor will be denoted byΔij.

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Lemma 3.13see1. A∈ AMzⁿif and only if for allg ∈RDandz /0,

Ag

z ⁿ

i1

αizg ωⁱ⁻¹z

, 3.37

whereαiz _n

j1Δijhj/z^j−1/Δn, for some{hj}ⁿ_j1inRD.

Proposition 3.14. For allz₀∈D,

{T^∗k_z₀ :T∈ AMzⁿ} zⁿ−zⁿ₀RD^⊥. Proof. ByLemma 3.6,

{T^∗k_z₀ :T ∈ AMzⁿ} ⊂zⁿ−zⁿ₀RD^⊥. Now we prove thatzⁿ− zⁿ₀RD^⊥⊂

{T^∗k_z₀ :T∈ AMzⁿ}.

Since

zⁿ−zⁿ₀ z−z0z−ωz0· · ·

z−ωⁿ⁻¹z0

, 3.38

byLemma 3.12,

zⁿ−zⁿ₀

RD ^⊥kerM^∗_zn−zⁿ0 !"

kz0, kωz0, . . . , k_ωⁿ⁻¹_z₀#

. 3.39

Setf a₁k_z₀a₂k_ωz₀· · ·a_nk_ωn−1z0 ∈kerM^∗_zn−zⁿ₀ witha₁, a₂, . . . , a_n ∈C. For allg ∈RD, we define an operatorT:RD→RDas follows:

Tgz a₁gz a₂gωz · · ·a_ng ωⁿ⁻¹z

. 3.40

ByLemma 3.13,T ∈ AMzⁿ. For allλ∈D,

T^∗k_z₀λ T^∗k_z₀, k_λkz0, Tk_λ &

k_z₀, a₁k_λz a₂k_λωz · · ·a_nk_λ

ωⁿ⁻¹z' a1kλz, kz0a2kλωz, kz0· · ·an

kλ ωⁿ⁻¹z , kz0

a1kλz0 a2kλωz0 · · ·ankλ ωⁿ⁻¹z0

.

3.41

For 0≤i≤n−1,

k_λωⁱz₀ ∞ m0

β²_m λ_m

ωⁱz₀^m

^∞

m0

β²_m ωⁱz₀_m

λ^m k_ωⁱ_z₀λ.

3.42