On Some Bergman Shift Operators

Complex Anal. Oper. Theory (2012) 6:829–842DOI 10.1007/s11785-010-0101-6

Complex Analysisand Operator Theory

On Some Bergman Shift Operators

Olof Giselsson · Anders Olofsson

Received: 25 January 2010 / Accepted: 7 July 2010 / Published online: 27 July 2010© Springer Basel AG 2010

Abstract An operator identity satisfied by the shift operator in a class of standardweighted Bergman spaces is studied. We show that subject to a pureness conditionthis operator identity characterizes the associated Bergman shift operator up to unitaryequivalence allowing for a general multiplicity. The analysis of the general case makescontact with the class of n-isometries studied by Agler and Stankus.

Keywords Bergman shift operator · Wold decomposition · Invariant subspace ·n-Isometry

Mathematics Subject Classification (2000) Primary 47B32;Secondary 47A15 · 46E22

0 Introduction

Let E be an auxiliary Hilbert space and let n ∈ Z+ be a positive integer. We denote

by An(E) the Bergman space of all E-valued analytic functions

Communicated by Daniel Aron Alpay.

Research of A. Olofsson was supported by GS Magnuson’s fund of the Royal Swedish Academy ofSciences.

O. Giselsson · A. Olofsson (B)Department of Mathematics, Faculty of Science, Centre for Mathematical Sciences,Lund University, P.O. Box 118, 221 00 Lund, Swedene-mail: [email protected]

O. Giselssone-mail: [email protected]

830 O. Giselsson, A. Olofsson

f (z) =∑

k≥0

ak zk, z ∈ D,

in the unit disc D with finite norm

‖ f ‖2An

=∑

k≥0

‖ak‖2μn;k < +∞,

where μn;k = 1/(k+n−1

k

)for k ≥ 0. The norm of An(E) can also be described as the

limit

‖ f ‖2An

= limr→1

∫

D̄

‖ f (r z)‖2 dμn(z),

where dμ1 is normalized Lebesgue arc length measure on the unit circle T = ∂D andfor n ≥ 2 the measure dμn is the normalized weighted area element given by

dμn(z) = (n − 1)(1 − |z|2)n−2 dA(z), z ∈ D.

Here dA is usual Lebesgue area measure normalized so that the unit disc has unitarea. Notice that A1(E) is the standard Hardy space and that A2(E) is the unweightedBergman space on the unit disc D. A standard reference for Bergman space theory onthe unit disc is Hedenmalm et al. [13].

The Bergman shift operator is the operator Sn on An(E) given by multiplication bythe complex coordinate:

Sn f (z) = z f (z), z ∈ D,

for f ∈ An(E). It is straightforward to see that Sn is a bounded operator on An(E)

which is injective and has closed range. An interesting fact is that the operator Sn

satisfies the operator identity

(S∗n Sn)−1 =

n−1∑

k=0

(−1)k(

n

k + 1

)Sk

n S∗kn . (0.1)

Notice that for n = 1 formula (0.1) says that S1 is an isometry S∗1 S1 = I . The full

identity (0.1) in the scale of spaces An(E) seems first to have appeared in Olofsson [17,Section 1] where it was used in a calculation of operator-valued Bergman inner func-tions. For n = 2 a similar to (0.1) looking operator inequality has found spectacularapplications related to invariant subspaces in the unweighted Bergman space in workof Shimorin [20] and Hedenmalm et al. [12]. Formula (0.1) has also been used in thestudy of Bergman space Toeplitz operators with harmonic symbols by Louhichi andOlofsson [14].

On Some Bergman Shift Operators 831

The purpose of the present paper is to investigate the extent to what the operatoridentity (0.1) characterizes the Bergman shift operator. This analysis amounts to pro-vide the counterpart of the Wold decomposition of an isometry (see Halmos [10] orSz.-Nagy and Foias [21, Section I.1]) for operators satisfying identity (0.2) below. Wewish to mention here also that the topic of Wold decompositions for operators closeto isometries has attracted additional interest from its relation to a celebrated approx-imation result for invariant subspaces of the unweighted Bergman space by Alemanet al. [7] (see also [16,20]). Let us describe the contents of the present paper.

Let H be a general not necessarily separable Hilbert space and denote by L(H) thespace of all bounded linear operators on H. Let T ∈ L(H) be an operator which isinjective and has closed range. Assume also that

(T ∗T )−1 =n−1∑

k=0

(−1)k(

n

k + 1

)T k T ∗k in L(H). (0.2)

Notice that the operator (T ∗T )−1 exists in L(H) since T is injective and has closedrange. We show that the subspace

H0 =⋂

k≥0

T k(H)

of H is reducing for T . The restricted operator T |H0 is then characterized by theproperty that its adjoint (T |H0)

∗ in L(H0) is an invertible n-isometry in the sense ofAgler and Stankus [3–5] (see Sect. 2).

We then turn our attention to injective operators T ∈ L(H) with closed rangesatisfying (0.2) that are also pure in the sense that

⋂

k≥0

T k(H) = {0}. (0.3)

We show that such an operator T is unitarily equivalent to Sn on An(E) allowingfor a general multiplicity E . A byproduct of this analysis is that an injective operatorT ∈ L(H) with closed range satisfying (0.2) is pure in the above sense of (0.3) if andonly if it has the property that T ∗k → 0 as k → ∞ in the strong operator topology (seeSect. 3). In closing the paper we discuss some applications of the preceding materialto shift invariant subspaces of An(E) (see Sect. 4).

1 Injective Operators With Closed Range

The purpose of this section is to recall some constructions related to an injective oper-ator with closed range. The setup is collected from [7,19,20]. Let T ∈ L(H) be aninjective operator with closed range. Equivalently this means that T ∈ L(H) is left-invertible. Yet another equivalent property is that the operator T ∈ L(H) is boundedfrom below in the sense that ‖T x‖2 ≥ c‖x‖2, x ∈ H, for some constant c > 0.


Two (closed) subspaces of H naturally associated to a left-invertible operatorT ∈ L(H) are

H0 =⋂

k≥0

T k(H) and E = H � T (H). (1.1)

The subspace H0 is invariant for T in the usual sense that T (H0) ⊂ H0. The subspaceE has the property that T k(E) ⊥ E for k ≥ 1 and is called the wandering subspace forT ∈ L(H). The terminology here originates from Halmos [10].

The operator

L = (T ∗T )−1T ∗ in L(H)

is the left-inverse of T with kernel E , that is, LT = I in L(H) and ker L = E . Theoperator

P = I − T L in L(H)

is the orthogonal projection of H onto E . Indeed, the operator T L is self-adjoint,idempotent and has range equal to T (H).

For x ∈ H we consider the E-valued analytic function V x defined by the formula

V x(z) = P(I − zL)−1x =∑

k≥0

(P Lk x)zk . (1.2)

It is straightforward to see that the map V : x �→ V x given by (1.2) intertwines theoperators T and L with the shift and backward shift operators in the sense that

V T x(z) = zV x(z) and V L(z) = (V x(z) − V x(0))/z,

respectively.

Lemma 1.1 Let T ∈ L(H) be an injective operator with closed range. Then

ker V ={

x ∈ H : P Lk x = 0 for all k ≥ 0}

= H0,

where H0 is as in (1.1).

Proof See Shimorin [20, Lemma 2.2]. � The operator

T ′ = T (T ∗T )−1 in L(H)

is called the (Cauchy) dual operator for T ∈ L(H). Notice that T ′ is left-invertibleand that the wandering subspaces for T and T ′ are the same, that is, E = E ′, whereE ′ = H � T ′(H). It is straightforward to see that the left-inverse of T ′ with kernel Eis T ∗, that is, L ′ = T ∗, where L ′ = (T ′∗T ′)−1T ′∗.


2 An Invertible Part

Let n ∈ Z+ be a positive integer. Following Agler and Stankus [3–5] an operator

T ∈ L(H) is called an n-isometry if

n∑

k=0

(−1)k(

n

k

)T ∗k T k = 0 in L(H).

Notice that this defining property of an n-isometry equivalently means that

n∑

k=0

(−1)k(

n

k

)‖T k x‖2 = 0, x ∈ H.

By this last equality we have that an operator T ∈ L(H) is an n-isometry if and onlyif for every x ∈ H the function N � k �→ ‖T k x‖2 is given by a polynomial in k ofdegree at most n − 1; here N is the set of non-negative integers.

Proposition 2.1 Let T ∈ L(H) be an invertible operator. Then T satisfies (0.2) if andonly if its adjoint T ∗ is an n-isometry.

Proof Since T is invertible, the identity (0.2) can be restated as

n−1∑

k=0

(−1)k(

n

k + 1

)T k T ∗k = T −1(T ∗)−1 in L(H).

Now multiplying to the left by T and to the right by T ∗ we see that (0.2) is equivalentto

n−1∑

k=0

(−1)k(

n

k + 1

)T k+1T ∗(k+1) = I,

which by some algebra simplifies to

n∑

k=0

(−1)k(

n

k

)T k T ∗k = 0 in L(H).

This last equality says that T ∗ is an n-isometry. � It is known that an invertible n-isometry is necessarily unitary for n = 1, 2 (see

[3, Proposition 1.23], [15, Proposition 1.1], [16, Section 2] or [20, Proposition 3.2]).For n ≥ 3 there are plenty of invertible n-isometries that are not unitary as has beenpointed out by Agler and Stankus [3, Section 3]. Notice that if T ∈ L(H) is an invert-ible n-isometry and H0 is a doubly invariant subspace for T meaning that H0 is aclosed subspace of H invariant for both T and T −1, then the restricted operator T |H0

in L(H0) is an invertible n-isometry. Let us give an example.


Let ω = {ωk}∞k=−∞ be a doubly infinite weight sequence such that the functionZ � k �→ ωk is strictly positive and given by a polynomial of degree at most n − 1.Denote by �2

ω the space of all sequences a = {ak}∞k=−∞ of complex numbers such that

‖a‖2ω =

∞∑

k=−∞|ak |2ωk < +∞.

Then the shift Sa = {ak−1}∞k=−∞ acting on �2ω is an invertible n-isometry. Indeed, it

is straightforward to see that the function Z � k �→ ‖Ska‖2ω is given by a polynomial

of degree at most n − 1.A special case of this construction is the shift operator S f (eiθ ) = eiθ f (eiθ ), eiθ ∈

T, acting on the Sobolev space W 1,2(T) which is an invertible 3-isometry obtainedfrom the weight sequence ωk = k2 + 1, k ∈ Z, as above. Observe that the functionsin W 1,2(T) are pointwise defined continuous functions on T. A natural subspace ofW 1,2(T) is the subspace IF consisting of all functions in W 1,2(T) vanishing on aclosed set F ⊂ T, that is,

IF ={

f ∈ W 1,2(T) : f (eiθ ) = 0 for all eiθ ∈ F}

.

It is evident that such a subspace IF is doubly shift invariant. Moreover, it is knownthat spectral synthesis holds in W 1,2(T) in the sense that every doubly shift invariantsubspace of W 1,2(T) has the form IF for some closed subset F of T (see [8,11]). Wemention that spectral synthesis problems in so-called regular rings have been studiedby Shilov and others (see [9, Chapter VI]).

We mention also that the class of 2-isometries has been much studied because ofits relation to the Dirichlet shift (see for instance [6,15,19]).

We shall need the following property of binomial coefficients.

Lemma 2.1 Let μn;k = 1/(k+n−1

k

)for n ≥ 1 and k ≥ 0. Then

min(n−1,k)∑

j=0

(−1) j(

n

j + 1

)1

μn;k− j= 1

μn;k+1.

Proof See [17, Lemma 1.1]. � Recall from Sect. 1 that L = (T ∗T )−1T ∗ is the left-inverse of T with kernel E

given by (1.1) and that P is the orthogonal projection of H onto E .

Lemma 2.2 Let T ∈ L(H) be a left-invertible operator satisfying (0.2). Then

P Lm = 1

μn;mPT ∗m in L(H) (2.1)

for every integer m ≥ 0.


Proof We shall prove (2.1) by induction on m. For m = 0 there is nothing to prove.Let m0 ≥ 1 be an integer and assume that formula (2.1) holds true for every integer

m in the range 0 ≤ m < m0. We proceed to show that then formula (2.1) holds form = m0 also. Using (0.2) we have that

P Lm0 = P Lm0−1(T ∗T )−1T ∗ = P Lm0−1

(n−1∑

k=0

(−1)k(

n

k + 1

)T k T ∗k

)T ∗.

Notice that for k ≥ m0 we have that P Lm0−1T k = 0 since LT = I and PT = 0. Bythis we see that

P Lm0 = P Lm0−1

⎛

⎝min(n−1,m0−1)∑

k=0

(−1)k(

n

k + 1

)T k T ∗k

⎞

⎠ T ∗.

Again using that LT = I we have that

P Lm0 =⎛

⎝min(n−1,m0−1)∑

k=0

(−1)k(

n

k + 1

)P Lm0−k−1T ∗k

⎞

⎠ T ∗.

We now use the induction hypothesis (2.1) for 0 ≤ m < m0 to conclude that

P Lm0 =⎛

⎝min(n−1,m0−1)∑

k=0

(−1)k(

n

k + 1

)1

μn;m0−k−1PT ∗(m0−k−1)T ∗k

⎞

⎠ T ∗

=⎛

⎝min(n−1,m0−1)∑

k=0

(−1)k(

n

k + 1

)1

μn;m0−k−1

⎞

⎠ PT ∗m0 = 1

μn;m0

PT ∗m0 ,

where the last equality follows by Lemma 2.1. By induction this completes the proofof the lemma. �

We are now ready for the main result of this section.

Theorem 2.1 Let T ∈ L(H) be an injective operator with closed range satisfying(0.2). Then the subspace

H0 =⋂

k≥0

T k(H)

of H is reducing for T and the restricted operator T ∗|H0 in L(H0) is an invertiblen-isometry.


Proof We shall first prove that

H0 =⋂

k≥0

T ′kH, (2.2)

where the operator

T ′ = T (T ∗T )−1 in L(H)

is the Cauchy dual of T . By this identity (2.2) it clearly follows that T ∗(H0) ⊂ H0,which gives that H0 is reducing for T .

Let us now turn to the proof of (2.2). By Lemma 1.1 we have that x ∈ H0 if andonly if P Lk x = 0 for k ≥ 0. Also by Lemma 2.2 we have that P Lk x = 0 if andonly if PT ∗k x = 0. Recall that T ∗ is the left-inverse of T ′ with kernel E . Anotherapplication of Lemma 1.1 now to the dual operator T ′ gives (2.2).

It is straightforward to see that the restricted operator T |H0 in L(H0) is invert-ible. By Proposition 2.1 we have that the adjoint operator (T |H0)

∗ = T ∗|H0 is ann-isometry. This completes the proof of the theorem. �

3 Pure Operators

In this section we shall study left-invertible operators T ∈ L(H) satisfying (0.2) thatare pure in the sense that

⋂

k≥0

T k(H) = {0}.

We shall show that the conditions of pureness and (0.2) canonically characterize theBergman shift Sn acting on An(E) allowing for a general multiplicity E .

Recall that the symbol∨

is commonly used to denote a closed linear span.

Lemma 3.1 Let T ∈ L(H) be a pure left-invertible operator satisfying (0.2). Then

H =∨

k≥0

T k(E),

where E = H � T (H) is the wandering subspace for T .

Proof Assume that x ∈ H is such that x ⊥ T k(E) for k ≥ 0. We shall show that thenx = 0. For y ∈ E we have

0 = 〈x, T k y〉 = 〈T ∗k x, y〉 = 〈PT ∗k x, y〉,

which gives that PT ∗k x = 0 for k ≥ 0. By Lemma 2.2 we conclude that P Lk x = 0for k ≥ 0. Now an application of Lemma 1.1 gives that x = 0. �


We can now model pure left-invertible operators satisfying (0.2) using the opera-tor Sn .

Theorem 3.1 Let T ∈ L(H) be an injective operator with closed range which is pureand satisfies (0.2). Then the map

V : H � x �→ V x

given by (1.2) is an isometry mapping H onto An(E) which intertwines the operatorsT and Sn in the sense that V T = Sn V .

Proof We consider first the case of an element x ∈ H which is a finite sum of theform x = ∑

k≥0 T k xk , where xk ∈ E . A straightforward calculation gives

V x(z) =∑

k≥0

xk zk

so that

‖V x‖2An

=∑

k≥0

‖xk‖2μn;k . (3.1)

We proceed to calculate ‖x‖2. Clearly

‖x‖2 =∑

j,k≥0

〈T j x j , T k xk〉 =∑

j,k≥0

〈PT ∗k T j x j , xk〉.

Now use Lemma 2.2 to conclude that

‖x‖2 =∑

j,k≥0

μn;k〈P Lk T j x j , xk〉.

Next notice that P Lk T j |E = 0 for j �= k. By this we see that

‖x‖2 =∑

k≥0

‖xk‖2μn;k . (3.2)

By (3.1) and (3.2) we have that ‖V x‖2An

= ‖x‖2 whenever x ∈ H is a finite sum of

the form x = ∑k≥0 T k xk with xk ∈ E for k ≥ 0.

By an approximation argument using Lemma 3.1 it now follows that the map V isan isometry of H into An(E). Since the set of E-valued polynomials in An(E) is densein An(E) it follows that V maps H onto An(E), that is, V (H) = An(E). It remains toshow that the map V intertwines the operators T and Sn . For x ∈ H we have that


V T x(z) =∑

k≥0

(P Lk T x)zk =∑

k≥1

(P Lk−1x)zk = zV x(z), z ∈ D.

This completes the proof of the theorem. � Remark 3.1 Let Tj ∈ L(H j ) be an operator satisfying the assumptions of the operatorT in Theorem 3.1 for j = 1, 2. It follows by Theorem 3.1 that the operators T1 andT2 are unitarily equivalent if and only if the wandering subspaces of T1 and T2 haveequal dimension.

We mention that for n = 2 the result of Theorem 3.1 is implicitly contained inShimorin [20]; see Remark following Corollary 3.7 in [20].

We say that an operator T ∈ L(H) belongs to the class C0· if limk→∞ T k = 0 in thestrong operator topology in L(H) meaning that T k x → 0 in H for every x ∈ H. Noticethat this terminology differ slightly from that of Sz.-Nagy and Foias [21, Section II.4].

Theorem 3.2 Let T ∈ L(H) be an injective operator with closed range satisfying(0.2). Then T is pure if and only if T ∗ belongs to the class C0·

Proof Assume first that T is pure. By Theorem 3.1 the operator T is unitarily equiv-alent to Sn acting on An(E) allowing for general multiplicity E . It is well-knownthat S∗k

n → 0 as k → ∞ in the strong operator topology (see for instance [18,Proposition 5.1]). This gives that T ∗ belongs to the class C0·

Assume next that T ∗ belongs to the class C0·. Let H0 be as in (1.1) and set T0 =T |H0 . By Theorem 2.1 the subspace H0 is reducing for T and T ∗

0 ∈ L(H0) is an invert-ible n-isometry. Let x ∈ H0. Since T ∗

0 is an n-isometry the function Z � k �→ ‖T ∗k0 x‖2

is given by a polynomial of degree at most n − 1 (see first paragraph in Sect. 2). Alsosince T ∗ belongs to the class C0· we have that ‖T ∗k

0 x‖2 → 0 as k → ∞. This forcesthat x = 0. It follows that H0 = {0}, which means that T is pure. �

Observe that the conditions of pureness and containment in the class C0· stay invari-ant under similarity of operators. As a result Theorem 3.2 generalizes accordingly.

Let T ∈ L(H) be a left-invertible operator satisfying (0.2). A calculation gives that

P = I − T (T ∗T )−1T ∗ =n∑

k=0

(−1)k(

n

k

)T k T ∗k in L(H).

By Lemma 2.2 we have that the map V given by (1.2) takes the form

V x(z) = Dn,T ∗(I − zT ∗)−n x, z ∈ D,

where

Dn,T ∗ =(

n∑

k=0

(−1)k(

n

k

)T k T ∗k

)1/2

in L(H)


using the positive square root. This shows that the map V has the form of a corre-sponding map used in the operator model theory for n-hypercontractions from [18,Sections 6–7] building on earlier work by Agler [1,2]. In fact the essence of the operatormodel theory for n-hypercontractions is to describe the parts of the adjoint shift S∗

n .

4 Invariant Subspaces of An(E)

In this section we shall specialize the preceding material to the context of invariantsubspaces of An(E). By an invariant subspace of An(E) we mean a closed subspaceI of An(E) such that Sn(I) ⊂ I.

Let E∗ and E be auxiliary Hilbert spaces. By a multiplier from An(E∗) into An(E)

we mean a bounded linear operator W from An(E∗) into An(E) commuting with theshift in the sense that W Sn = SnW .

Theorem 4.1 Let I be an invariant subspace of An(E), where n ∈ Z+ is a positive

integer. Then the following assertions are equivalent:

(1) The operator T = Sn|I satisfies (0.2).(2) The operator T = Sn|I is unitarily equivalent to Sn acting on some space An(E∗).(3) The subspace I has the form

I = W An(E∗) = {W f : f ∈ An(E∗)}

for some isometric multiplier W from An(E∗) into An(E).

Proof The implication (3) implies (2) is obvious since the multiplier W provides aunitary equivalence between the operators T and Sn acting on An(E∗). By the resultof [17, Section 1] we know that the shift operator T = Sn satisfies (0.2). This makesevident the implication (2) implies (1). It remains to show that (1) implies (3).

Assume that assertion (1) holds. By Theorem 3.1 there exists an isometry V map-ping I onto An(E∗) such that V T = Sn V , where T = Sn|I . Set W = V −1. Then Wmaps An(E∗) isometrically into An(E) and commutes with the shift. We conclude thatW is an isometric multiplier from An(E∗) into An(E) such that I = W An(E∗). �

We shall next consider the multipliers in Theorem 4.1 in some more detail. Thefollowing result is known but included here for the sake of convenience.

Proposition 4.1 A bounded operator W : An(E∗) → An(E) is a multiplier if andonly if it acts as

W f (z) = W̃ (z) f (z), z ∈ D, (4.1)

on functions f ∈ An(E∗), where W̃ is a bounded L(E∗, E)-valued analytic functionin D. Furthermore, the operator norm of the multiplier W is given by

‖W‖ = supz∈D

‖W̃ (z)‖

when W and W̃ are related by (4.1)


Sketch of proof It is straightforward to see using the integral formula for the norms ofAn(E∗) and An(E) from Sect. 0 that every operator W of the form (4.1) is a multiplierfrom An(E∗) into An(E) with ‖W‖ ≤ supz∈D‖W̃ (z)‖.

Assume next that W is a multiplier from An(E∗) into An(E). For a ∈ E∗, thefunction Wa in An(E) is given by a power series

Wa(z) =∑

k≥0

bk zk, z ∈ D,

with coefficients bk ∈ E for k ≥ 0. We now define operators Wk ∈ L(E∗, E) by settingWka = bk for k ≥ 0. It is straightforward to see that ‖Wk‖2 ≤ 1/μn;k‖W‖2 for k ≥ 0.This provides us with an L(E∗, E)-valued analytic function given by the power series

W̃ (z) =∑

k≥0

Wk zk, z ∈ D.

For f = ∑k≥0 Sk

n ak a polynomial in An(E∗) we have W f = ∑k≥0 Sk

n Wak , and apoint evaluation gives that

W f (z) =∑

k≥0

zk

⎛

⎝∑

j≥0

W j ak z j

⎞

⎠ = W̃ (z) f (z), z ∈ D.

We now have (4.1) for f ∈ An(E∗) a polynomial. The general case of (4.1) followsby approximation.

It remains to prove the inequality supz∈D‖W̃ (z)‖ ≤ ‖W‖. We shall employ thereproducing kernel function

Kn(z, ζ ) = 1

(1 − ζ̄ z)n, (z, ζ ) ∈ D × D,

for the space An which is characterized by the reproducing property that

〈 f (ζ ), b〉 = 〈 f, K (·, ζ )b〉An , ζ ∈ D,

for f ∈ An(E) and b ∈ E (see for instance [13, Section 1.1]). For a ∈ E∗ and b ∈ Ewe have

Kn(ζ, ζ )〈W̃ (ζ )a, b〉 = 〈W Kn(·, ζ )a, Kn(·, ζ )b〉An ,

and an application of the Cauchy-Schwarz inequality gives that

Kn(ζ, ζ )|〈W̃ (ζ )a, b〉| ≤ ‖W Kn(·, ζ )a‖An ‖Kn(·, ζ )b‖An ≤ Kn(ζ, ζ )‖W‖‖a‖‖b‖.

Canceling the factor Kn(ζ, ζ ) and taking an appropriate supremum we obtain theinequality that supζ∈D‖W̃ (ζ )‖ ≤ ‖W‖. �


It is natural not to distinguish too strict between multipliers from An(E∗) into An(E)

and bounded L(E∗, E)-valued analytic functions in D. We shall next consider isometricmultipliers.

Isometric multipliers from the Hardy space A1(E∗) into the Hardy space A1(E)

are usually called L(E∗, E)-valued inner functions. A bounded L(E∗, E)-valued ana-lytic function in D is an inner function if and only if its boundary value W (eiθ ) is anisometry in L(E∗, E) for a.e. eiθ ∈ T (see [21, Section V.2] for details).

Proposition 4.2 Let n ≥ 2. An L(E∗, E)-valued analytic function W in D is an iso-metric multiplier from An(E∗) into An(E) if and only if it has the form

W (z) = W0, z ∈ D,

for some isometry W0 ∈ L(E∗, E).

Proof The if-part is obvious. We proceed to prove the only if-part. Assume that W isan isometric multiplier and consider the power series expansion

W (z) =∑

k≥0

Wk zk, z ∈ D,

where the coefficients Wk are operators in L(E∗, E) for k ≥ 0. We shall show that W0is an isometry and Wk = 0 for k ≥ 1. Let a ∈ E∗. Using properties of W we have theequations that

‖a‖2μn;m = ‖Smn a‖2

An= ‖W Sm

n a‖2An

= ‖Smn Wa‖2

An=

∑

k≥0

‖Wka‖2μn;m+k .

(4.2)

for m = 0, 1, 2, . . .. Observe that the weight sequence {μn;k}k≥0 is decreasing andregular as k → ∞ since 1/μn;k is a polynomial in k of degree n −1 with non-negativecoefficients (see for instance [18, Section 5]). Dividing in (4.2) by μn;m we obtain

‖a‖2 =∑

k≥0

‖Wka‖2μn;m+k/μn;m,

and by a passage to the limit as m → ∞ using dominated convergence we see that

‖a‖2 =∑

k≥0

‖Wka‖2. (4.3)

Also setting m = 0 in (4.2) we have that

‖a‖2 = ‖Wa‖2An

=∑

k≥0

‖Wka‖2μn;k . (4.4)


Since 1 = μn;0 > μn;k for k ≥ 1, we conclude from (4.3) and (4.4) that ‖W0a‖2 =‖a‖2 and ‖Wka‖2 = 0 for k ≥ 1. This completes the proof of the proposition. � Remark 4.1 We remark that for n ≥ 2 the assertion (3) of Theorem 4.1 is equivalentto the assertion that:

(3’) The subspace I of An(E) has the form I = An(E∗) for some closed subspaceE∗ of E .

Indeed, this follows by Proposition 4.2.

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On Some Bergman Shift Operators

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