THE ESSENTIAL NORMS OF COMPOSITION OPERATORS

Boo Rim Choe

Abstract. Let ϕ be a holomorphic function that maps the unit ball of Cn into the unit disc.It is shown that the essential norm of the composition operator between the Hardy spacesinduced by ϕ is exactly the same as that of the inclusion operator induced by the pullbackmeasure of ϕ and is comparable with the essential Carleson norm of the pullback measure ofϕ. As an application the essential norms of composition operators induced by certain classesof inducing functions are explicitly calculated: two classes are considered. The anlogues inhigher dimensional cases and in the Bergman context are indicated without detail.

1. Introduction

Throughout the paper n denotes a fixed positive integer unless otherwise specified. LetB = Bn denote the open unit ball of Cn and let S = Sn denote its boundary, the unitsphere. The unique rotation-invariant probability measure on S will be denoted by σ = σn.For n = 1, we use more custumary notations D = B1, T = S1 and dσ1 = dθ/2π. TheHardy space on B, denoted by H2(B), is then the space of functions f holomorphic on Bfor which

||f ||2 = sup0<r<1

∫

S

|f(rζ)|2dσ(ζ) <∞.

Let ϕ be a holomorphic function on B such that ϕ(B) ⊂ D. For a function f holo-morphic on D it is clear that the composite function f ϕ is holomorphic on B. If, inaddition, f ∈ H2(D), then f ϕ ∈ H2(B). This is a consequence of the harmonic majorantcharacterization of the Hardy spaces (see [13, section 5.6]): a function g holomorphic on Bbelongs to H2(B) if and only if |g|2 has a harmonic majorant. The composition operatorCϕ : H2(D) −→ H2(B), Cϕf = f ϕ, induced by ϕ is therefore bounded by the closedgraph theorem.

In case B = D (i.e. n = 1) relations between the operator theoretic properties of Cϕand the function theoretic properties of the inducing function ϕ have been extensivelystudied by many authors. See, for example, [5], [16] and references therein. In particular,J. Shapiro [16] has recently completely answered the question “when is Cϕ compact?” Infact he obtained a complete function theoretic description of the essential norm (=distanceto the compact operators) of Cϕ in terms of the Nevannlina counting function of ϕ.

In the present paper we investigate the same question for general Cϕ in terms of thepullback measures τϕ on D defined by τϕ(E) = σ[(ϕ∗)−1(E)] (the superscript * denotes

1991 Mathematics Subject Classification. Primary 47B38, 32A35. Secondary 30D55.The author was in part supported by the Korea Science and Engineering Foundation

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the radial limit) for Borel subsets E of D. By a standard approximation one can easilyfind

(1)∫

S

f ϕ∗dσ =∫

D

f dτϕ

for every Borel function f ≥ 0 on D. Note that (f ϕ)∗ = f ϕ∗ [σ] a.e. on S for functionsf continuous on D. Thus it is easy to see ||Cϕ|| = ||Iτϕ || where Iτϕ denotes the denselydefined inclusion operator of H2(D) into L2(τϕ). In fact the following is true:

(2) ||Cϕ||2 = ||Iτϕ ||2 ∼= ||τϕ||C

where ||τϕ||C denotes the Carleson norm of τϕ (see Section 2). The notation ∼= means thatthe ratios of two terms are bounded below and above by constants Cn. Here, and elsewhere,Cn denotes an absolute constant, depending on the dimension n, which may change witheach occurrence. The second half of (2) is well known for measures τϕ concentrated onD, as the Carleson measure theorem, and remains true for measures τϕ not necessarilyconcentrated on D (see 3.3). In Section 3 we obtain a complete analogue of (2) for essentialnorms. In Section 4 we apply this result to calculate the precise essential norms of certaincomposition operators. Two cases are considered here: the first one involves inner functionsand the second one deals with the so-called orthogonal functions. Finally in Section 5 weindicate without detail the analogues in higher dimensional cases and in the Bergmancontext.

2. Preliminary

For a function f : B −→ C, we let f∗(ζ) = limr1 f(rζ) if the limit exists at ζ ∈ S.It is a well-known fact that if f ∈ H2(B), then f∗ exists for [σ] a.e. on S and the mapf −→ f∗ is a linear isometry of H2(B) into L2(σ). We will often use the same letter for afunction f ∈ H2(B) and its boundary function f ∈ L2(σ).

For α ∈ B we shall let Pα denote the test function defined by

Pα(z) =( √1− |α|2

1− < z, α >

)n(z ∈ B).

Here < , > denotes the complex inner product on Cn. By the Cauchy formula on B wehave Pα ∈ H2(B) and ||Pα|| = 1. The Poisson-Szego kernel on B is the kernel

P (z, ζ) =(1− |z|2)n

|1− < z, ζ > |2n (z ∈ B, ζ ∈ S).

Note that P (z, ζ) = |Pz(ζ)|2 for every test function Pz. The Poisson-Szego integral P [λ]of a complex Borel measure λ on S is defined by

P [λ](z) =∫

S

P (z, ζ) dλ(ζ) (z ∈ B).

2

If dλ = fdσ, then we write P [λ] = P [f ]. The symmetric derivative Dλ of a complex Borelmeasure λ on S is defined for [σ] a.e. ζ ∈ S by

Dλ(ζ) = limt0

λ(Q(ζ, t))σ(Q(ζ, t))

whereQ(ζ, t) = η ∈ S : |1− < η, ζ > | < t.

By Koranyi’s theorem we have

(1) P [λ]∗ = Dλ [σ] a.e. .

This happens at every Lebesgue point of f if dλ = fdσ. Thus dλ = Dλdσ for λ << σ.See [13, Section 5.4] for details.

In what follows the term “measure” will always refer to positive finite Borel measures.Let τ be a measure on B. We shall let Iτ denote the densely defined “inclusion” operatorof the Hardy space H2(B) into L2(τ). We also define its “Carleson norm” (see 3.3 below)

||τ ||C = supτ(Ω(ζ, t))σ(Q(ζ, t))

(ζ ∈ S, t > 0)

and its “essential Carleson norm”

|||τ |||C = lim supt0

supζ∈S

τ(Ω(ζ, t))σ(Q(ζ, t))

whereΩ(ζ, t) = z ∈ B : |1− < z, ζ > | < t.

A measure on a Borel subset E of B will be considered as a measure on B concentratedon E. For example, λ(Ω(ζ, t)) = λ(Q(ζ, t)) for a measure λ on S.

Given a densely defined linear operator L of a Hilbert space X into another Hilbertspace Y , we will denote the operator norm of L by ||L|| when there is no possibility ofconfusion with the H2 norm. If ||L|| < ∞, we will use the same letter L for the uniquebounded extension on all of X. The essential norm |||L||| of bounded L is its distance tothe compact operators. More precisely,

|||L||| = inf||L−K|| : K a compact operator of X into Y .

We will not complicate the notation by indicating the spaces explicitly in ||L|| and |||L|||.This should cause no confusion. Finally, dimensions involved in various notations definedabove will be clear from the context.

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3. Essential norms

Let ϕ : B −→ D be a holomorphic function and let τϕ be its pullback measure onD. Since Cϕ is bounded, 1.(1) and a consideration of test functions therefore easily yield||τ ||C < ∞. The same is true for the restriction of τϕ to T . With this in mind we provethe following.

3.1. Proposition. Let λ be a measure on S. Then the following conditions are equiva-lent.

(i) ||λ||C <∞;(ii) λ << σ and Dλ ∈ L∞(σ);

(iii) Iλ is bounded.If one of the above holds, then

(1) |||Iλ|||2 = ||Iλ||2 = ||Dλ||∞ ∼= ||λ||C .

Proof. (i) =⇒ (ii): Let Mλ be the maximal function of the measure λ defined by

Mλ(ζ) = supt>0

λ(Q(ζ, t))σ(Q(ζ, t))

(ζ ∈ S).

By assumption Mλ(ζ) ≤ ||λ||C <∞ for every ζ ∈ S and therefore, by [13, Theorem 5.2.7],λ << σ. Clearly, ||Dλ||∞ ≤ ||λ||C .

(ii) =⇒ (iii): Since λ << σ, we have dλ = Dλdσ. Consequently, ||Iλ||2 ≤ ||Dλ||∞ <∞.

(iii) =⇒ (i): Apply Iλ to test functions to derive ||Iλ||2 ≥ Cn||λ||C .Hence the conditions (i) ∼ (iii) are equivalent and, moreover, if one of those three

conditions holds, then the proof above shows that Cn||λ||C ≤ ||Iλ||2 ≤ ||Dλ||∞ ≤ ||λ||C .Thus, to prove (1), it remains to show that |||Iλ|||2 ≥ ||Dλ||∞.

Since lim|α|1 |h(α)|2(1−|α|2)n = 0 for every h ∈ H2(B) (see [13, Theorem 7.2.5]), thesequence of test functions Pα converges to 0 weakly on H2(B) as |α| 1 by the Cauchyformula. Let K : H2(B) −→ L2(λ) be a compact operator. Since K transforms weaklyconvergent sequences into norm convergent ones (see [19, Theorem 6.3]), we have

(1) lim|α|1

||KPα||2 = 0

where || ||2 denotes the L2(λ) norm. Write L = Iλ for notational simplicity. Then, forevery α ∈ B,

||L−K|| ≥ ||(L−K)Pα||2 ≥ ||LPα||2 − ||KPα||2,so by (1),

||L−K|| ≥ lim sup|α|1

||LPα||2.

Taking the infimum on the left side of the above over all K and writing α = (1 − t)ζ(ζ ∈ S, 0 < t < 1), we obtain

|||L|||2 ≥ lim supt0

supζ∈S

∫

S

|Pα|2dλ.

4

Thus we have|||L|||2 ≥ lim sup

t0P [λ]((1− t)ζ) (ζ ∈ S).

This completes the proof by 2.(1).

The following lemma is originally due to Ryff [15] in the setting B = D. A similarlemma appears in [8].

3.2. Lemma. Let ϕ : B −→ D be a holomorphic funcion and assume f ∈ H2(D). Then

(f ϕ)∗ = f ϕ∗ [σ] a.e.

where f = f on D and f = f∗ on T .

Proof. First of all, note that the lemma is true if f is continuous on D. Let fr (0 < r < 1)be the dilated function z −→ f(rz) on D. Then fr −→ f in H2(D) and thus

(1) limr1

∫

S

|(f ϕ)∗ − fr ϕ∗|2dσ = 0.

Let λ be the restriction of τϕ to T . Since ||λ||C <∞ as remarked earlier, f∗ exists [λ] a.e.on T by Proposition 3.1. From this fact one easily obtains

(2) limr1

fr ϕ∗(ζ) = f ϕ∗(ζ) for [σ] a.e. ζ ∈ S.

An application of Fatou’s lemma now shows the lemma by (1) and (2).

3.3. CARLESON MEASURES. A measure τ on B such that ||τ ||C < ∞ is called aCarleson measure. The importance of Carleson measures lies in the so-called Carlesonmeasure theorem ([1], [8]; also [4], [11]):

(1) ||Iτ ||2 ∼= ||τ ||C

holds for measures τ on B. What is needed here is a slight variation of this result, whereτ is not necessarily concentrated on B. In this setting the inequality ||Iτ ||2 ≥ Cn||τ ||Cfollows as before by considering test functions. For the other direction of the inequality,suppose ||τ ||C < ∞. Let µ and λ be the restrictions of τ to B and S, respectively. Thenit is easily verified from (1) and Proposition 3.1 that

||Iτ ||2 ≤ ||Iµ||2 + ||Iλ||2 ∼= ||µ||C + ||λ||C ≤ 2||τ ||C .

The following must have been also well known to some workers in the field. We includeit for the sake of completeness.

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3.4. Proposition. |||Iτ |||2 ∼= |||τ |||C holds for measures τ on B.

Proof. Write L = Iτ again for notational simplicity. Then, as in the proof of Proposition3.1, we have

|||L|||2 ≥ lim supt0

supζ∈S

∫

B

|P(1−t)ζ |2dτ.

Replace B by Ω(ζ, t) in the integral of the above. Then the resulting integral is easily seento be bounded below by Cnτ(Ω(ζ, t))t−n. Since (see [13, Proposition 5.1.4])

(1) σ(Q(ζ, t)) ∼= tn (0 < t < 2),

we obtain|||L|||2 ≥ Cn|||τ |||C .

Now we prove the other direction of the inequality. For 0 < r < 1, let Lr = Iτr whereτr denotes the restriction of τ to the annulus B \ (1 − r)B and let Mr : L2(τ) −→ L2(τ)be the multiplication by the characteristic function of (1− r)B. Then Mr L is compact(by a normal family argument). Thus

|||L||| ≤ ||L−Mr L|| = ||Lr||

and therefore, since the sequence ||Lr|| is monotone, we have

(2) |||L||| ≤ limr0||Lr||.

Hence the following inequality will finish the proof by (2) and 3.3.(1):

||τr||C ≤ Cn|||τ |||C + o(1) (r 0).

The above is implicit in the proof of Theorem 1.1 of [8] in case |||τ |||C = 0 and a slightmodification shows the general case. We include it for completeness. Fix ζ ∈ S. We needverify

(3) sup0<t<2

τr(Ω(ζ, t))σ(Q(ζ, t))

≤ Cn|||τ |||C + o(1) (uniformly in ζ).

Since τr(Ω(ζ, t)) = τ(Ω(ζ, t)) for t ≤ r,

(4) sup0<t≤r

τr(Ω(ζ, t))σ(Q(ζ, t))

≤ |||τ |||C + o(1) (uniformly in ζ).

Now assume r < t < 2. Before proceeding, we introduce a temporary notation:

D(ζ, δ) = z ∈ B : |z| > 1− δ, z/|z| ∈ Q(ζ, δ).

6

Note that Ω(ζ, δ) ⊂ D(ζ, δ) ⊂ Ω(ζ, 2δ). Let η1, η2, · · · , ηN be a maximal collection ofpoints in Q(ζ, t) subject to the condition |1− < ηi, ηj > | ≥ r for i 6= j. By the triangleinequality (see [13, Proposition 5.1.2.])

|1− < a, b > |1/2 ≤ |1− < a, c > |1/2 + |1− < b, c > |1/2 (a, b, c ∈ B),

the balls Q(ηj , r/4) are pairwise disjoint and contained in Q(ζ, 3t). Hence N ≤ Cn(t/r)n

by (1). On the other hand, the balls Q(ηj , r) cover Q(ζ, t) by maximality and thus

Ar ∩ Ω(ζ, t) ⊂ Ar ∩D(ζ, t) ⊂ ∪Nj=1D(ηj , r) ⊂ ∪Nj=1Ω(ηj , 2r).

Since N ≤ Cn(t/r)n, the above combined with (1) implies

supr<t<2

τr(Ω(ζ, t))σ(Q(ζ, t))

≤ Cn supη∈S

τ(Ω(η, 2r))σ(Q(η, 2r))

.

The right side of the above is dominated by Cn|||τ |||C + o(1) where o(1) is independent ofζ. This, together with (4), shows (3).

We are now ready to prove the main result of this section.

3.5. Theorem. Let ϕ : B −→ D be a holomorphic funcion. Then

|||Cϕ|||2 = |||Iτϕ |||2 ∼= |||τϕ|||C .

Proof. For simplicity write τ = τϕ. Recall that Iτ is bounded. Let R : H2(B) −→ L2(σ)be the isometry defined by Rf = f∗ and let W : L2(τ) −→ L2(σ) be the isometry definedby Wg = g ϕ∗. Then Lemma 3.2 is simply

(1) R Cϕ = W Iτ .Now let W ∗ : L2(σ) −→ L2(τ) be the adjoint of W defined by the requirement

∫

S

f ·Wg dσ =∫

D

W ∗f · g dτ.

It is easy to find from 1.(1) that W ∗ W is the identity on L2(τ). Also note that W ∗ isnorm-decreasing. Hence, for a compact operator K : H2(D) −→ H2(B), we obtain from(1) that

||Cϕ −K|| = ||W Iτ −R K|| ≥ ||Iτ −W ∗ R K|| ≥ |||Iτ |||.Upon taking infimum over all K, we find |||Cϕ||| ≥ |||Iτ |||. By using the Cauchy transformC : L2(σ) −→ H2(B) (see [13, Section 6.3]) one can prove

||Iτ −K|| ≥ |||Cϕ − C W K|| ≥ |||Cϕ|||in a similar way for any compact operator K : H2(D) −→ L2(τ) and thus |||Iτ ||| ≥ |||Cϕ|||.Consequently, |||Cϕ|||2 = |||Iτ |||2 ∼= |||τ |||C by Proposition 3.4. 3.6. Corollary. For a holomorphic function ϕ : B −→ D, Cϕ is compact if and only if|||τϕ|||C = 0.

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4. Two special cases

The explicit value of the norm or essential norm of Cϕ has been known only for veryspecial classes of inducing functions ϕ. See [5], [10] and [16]. In this section we calculatethe precise essential norms of composition operators induced by two special classes offunctions.

The first one involves inner functions. Recall that a holomorphic function ϕ : B −→ Dis called inner if |ϕ∗| = 1 [σ] a.e.. The norm of a composition operator Cϕ induced by aninner function ϕ on D was calculated by Nordgren ([10, Theorem 1]):

||Cϕ||2 =1 + |ϕ(0)|1− |ϕ(0)| .

Recently Shapiro ([16, Theorem 2.5]) has proved that such a composition operator hasthe same essential norm as above by using his remarkable essential norm formula ([16,Theorem 2.3]) for arbitrary composition operators on H2(D). It is an interesting factthat the (essential) norm formula for Cϕ induced by an inner function ϕ is independentof dimensions. This is contained in Theorem 4.2. Our derivation is independent. We firstprove a lemma.

4.1. Lemma. Suppose τ is a measure on B and let τr (0 < r < 1) be its restriction tothe annulus B \ (1− r)B. Then

|||Iτ ||| = limr0||Iτr ||.

Proof. Put L = Iτ and Lr = Iτr . Let K : H2(B) −→ L2(τ) be an arbitrary compactoperator and let Nr : L2(τ) −→ L2(τ) be the multiplication by the characteristic functionof B \ (1− r)B. Then, since ||Nr|| ≤ 1, we have

||L−K|| ≥ ||Nr L−Nr K|| ≥ ||Nr L|| − ||Nr K|| = ||Lr|| − ||Nr K||.

The sequence Nr is uniformly bounded and converges to 0 pointwise on L2(τ) as r 0.It is therefore equicontinuous and converges to 0 uniformly on compact subsets of L2(τ)by Ascoli’s theorem. In particular, this happens on K(Φ) where Φ is the closed unit ballof H2(B). This means limr0 ||Nr K|| = 0. It follows that ||L−K|| ≥ limr0 ||Lr|| andthus |||L||| ≥ limr0 ||Lr||. For the other direction, see 3.4.(2). 4.2. Theorem. Suppose ϕ is an inner function on B. For 0 < t ≤ 1, let ψ = (1− t)b+ tϕwhere b is a unimodular number such that b|ϕ(0)| = ϕ(0). Then

||Cψ||2 = |||Cψ|||2 =1t· 1 + |ϕ(0)|

1− |ϕ(0)| .

Proof. Let P (z, eiθ) be the Poisson kernel on D. Since ϕ is an inner function, we have([10], [12])

dτϕ(θ) = P (a, eiθ)dθ

2π(a = ϕ(0)).

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The pullback measure τ = τψ is therefore a Carleson measure on D such that

(1)∫

D

f dτ =∫ 2π

0

f h(eiθ)P (a, eiθ)dθ

2π

for every Borel function f ≥ 0 on D where h(z) = (1 − t)b + tz for z ∈ D. Hence thetheorem is a special case of the following result by Theorem 3.5.

We continue using the notation defined in the proof above.

4.3. Theorem. Put τ1 = τ and let τr (0 < r < 1) be the restriction of τ to the annulusD \ (1− r)D. Then

||Iτr ||2 = |||Iτr |||2 = |||τr|||C =1t· 1 + |a|

1− |a| (0 < r ≤ 1).

Note: This shows that the equality can happen in Proposition 3.4 in a nontrivial way.

Proof. Write Lr = Iτr . By Cowen’s norm formula ([5, Theorem 3]) we have ||Ch||2 = t−1.Hence it is clear from 4.2.(1) that

(1) ||Lr||2 ≤ ||Ch||2 · 1 + |a|1− |a| =

1t· 1 + |a|

1− |a| .

On the other hand, considering test functions on D, we have

(2) ||Lr||2 ≥∫ 2π

0

|Pα(h(eiθ))|2χr(eiθ)P (a, eiθ)dθ

2π(α ∈ D)

where χr denotes the characteristic function of the set of points eiθ such that |h(eiθ)| > r.For α = sb (0 < s < 1) a straightforward calculation shows

|Pα(h(eiθ))|2 =1 + s

1− s+ 2tsP( ts

1− s+ tsb, eiθ

).

Inserting the above into (2), we find

||Lr||2 ≥ 1 + s

1− s+ 2tsP [χr · P (a, ·)]

( ts

1− s+ tsb).

Take the limit s 1 on the right side of the above and use 2.(1) to obtain

(3) ||Lr||2 ≥ t−1χr(b)P (a, b) =1t· 1 + |a|

1− |a| .

Thus, by (1), (3) and Lemma 4.1,

||Lr||2 = |||Lr|||2 =1t· 1 + |a|

1− |a| .

9

It remains to compute |||τr|||C . Note that

supθτ(Ω(eiθ, s)) = τ(Ω(b, s))

and thath(eiθ) ∈ Ω(b, s) ⇐⇒ eiθ ∈ Qs/t.

Here Qδ = Q(b, δ) for δ > 0. Also note that limδ0 δ−1|Qδ| = π−1 where |Qδ| denotes the

normalized arclength of Qδ. It follows from these observations and 4.2.(1) that

|||τ |||C =1t

limδ0

1|Qδ|

∫

Qδ

P (a, eiθ)dθ

2π=

1t· 1 + |a|

1− |a| .

The second equality of the above holds by continuity of P (a, ·) at b. This completes theproof because |||τ |||C = |||τr|||C by definition.

We now turn to the so-called orthogonal functions. We shall use the notation

Eϕ = ζ ∈ S : ϕ∗(ζ) exists and is unimodular.

4.4. ORTHOGONAL FUNCTIONS. Let us say that a holomorphic function ϕ : B −→ Dis orthogonal if the sequence 1, ϕ, ϕ2, · · · forms an orthogonal set in H2(B). It is clearthat the rotation-invariance of τϕ implies the orthogonality of ϕ. It is less clear that theconverse is also true: the orthogonality of ϕ implies the rotation-invariance of τϕ. Thisfollows from the fact that if ϕ is orthogonal , then

∫

D

h(eiθz) dτϕ(z) =∫

D

h(z) dτϕ(z)

holds for θ real and functions h of the form h(z) = zkz` (k, ` ≥ 0) which span a densesubset of L1(τϕ). It is easy to see that if ϕ is an inner function on B with ϕ(0) = 0,then ϕ is orthogonal. Also, every nonconstant (holomorphic) homogeneous polynomialϕ : B −→ D is orthogonal. There are others: for n ≥ 2 one may construct orthogonalfunctions by using the integral formula

(1)∫

S

fdσ =(n− 1k

)∫

Bk

(1− |z|2)`−1dVk(z)∫

S`

f(z,√

1− |z|2ζ) dσ`(ζ)

where k + ` = n, f ∈ L1(σ), and Vk denotes the normalized volume measure on Bk.If k = n − 1, then (1) is Proposition 1.4.7 of [13]. This general form can be shown bythe “same” proof. Also, the orthogonality is invariant under compositions with measurepreserving maps ( [14, Chapter14], [18]). Moreover, the author has been informed thatJose Fernandez has constructed orthogonal functions ϕ, on D and hence on B, for which0 < σ(Eϕ) < 1.

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4.5. Lemma. If τ is a rotation-invariant measure on B, then |||τ |||C ≤ τ(S).

Proof. Since τ is rotation-invariant, we have

(1) τ(Ω(e1, t)) = τ(Ω(ζ, t)) (ζ ∈ S, t > 0).

Let At = B \ (1 − t)B (0 < t < 1). Then Ω(ζ, t) ⊂ At and thus, by (1) and Fubini’stheorem,

(2) τ(Ω(e1, t)) =∫

S

τ(Ω(ζ, t)) dσ(ζ) =∫

At

σ(E(z, t)) dτ(z)

where E(z, t) = ζ ∈ S : |1− < z, ζ > | < t. Note that E(z, t) ⊂ Q(z/|z|, t) and thusσ(E(z, t)) ≤ σ(Q(e1, t)) for every z ∈ At. Thus, by (2),

τ(Ω(e1, t)) ≤ τ(At)σ(Q(e1, t)).

This implies |||τ |||C ≤ limt0 τ(At) = τ(∩At) = τ(S). 4.6. Lemma. Suppose τ is a measure on B such that ||τ ||C < ∞. Let µ and λ be itsrestrictions to B and S, respectively.

(i) If |||µ|||C = 0, then |||Iτ |||2 = ||Dλ||∞ .(ii) If τ is rotation-invariant, then |||Iτ |||2 = τ(S).

Proof. By considering test functions as before, one can easily obtain |||Iτ |||2 ≥ ||Dλ||∞from 2.(1). Suppose |||µ|||C = 0 and let MB : L2(τ) −→ L2(τ) be the multiplication bythe characteristic function of B. Then one easily obtains |||MB Iτ ||| = |||Iµ|||. Thus, byProposition 3.4, MB Iτ is compact and hence |||Iτ ||| ≤ ||Iτ −MB Iτ || = ||Iλ|| so that(i) holds by Proposition 3.1.

Now assume τ is rotation-invariant. Then µ and λ are also rotation-invariant. Thus|||µ|||C = 0 by Lemma 4.5 and dλ = λ(S)dσ by the uniqueness of the rotation-invariantprobability measure on S. Accordingly, |||Iτ |||2 = ||Dλ||∞ = λ(S) = τ(S) by (i).

Note that τϕ(T ) = σ(Eϕ). Thus, combining Theorem 3.5 and Lemma 4.6, we obtain

4.7. Theorem. If ϕ is an orthogonal function on B, then |||Cϕ|||2 = σ(Eϕ). For n ≥ 2 there is a sufficient condition which implies σ(Eϕ) = 0 for holomorphic

functions ϕ : B −→ D. For references concerning this topic, see [14, Section 15.3]. Forexample, if ϕ is continuously differentiable up to the boundary, then σ(Eϕ) = 0. Such afunction therefore induces a compact composition operator whenever it is orthogonal byTheorem 4.7. In particular, every homogeneous polynomial that maps B onto D inducesa compact composition operator. This composition operator, however, is not always aHilbert-Schmidt operator as the following example shows. Note that Cϕ is a Hilbert-Schmidt operator if and only if (1− |ϕ|2)−1 ∈ L1(σ).

4.8. Example. Let ϕ : B2 −→ D be an orthogonal function, with ϕ(B2) = D, takingone of the following forms:

(i) ϕ(z1, z2) = zα11 (1− zα2

2 )−1 (α1, α2 ≥ 2);(ii) ϕ(z1, z2) = a1z

d1 + a2z

d2 ;

(iii) ϕ(z1, z2) = bzα11 zα2

2 .

11

Note that σ(Eϕ) = 0 and thus Cϕ is compact by Theorem 4.7. For such a function ϕ, wehave τϕ << m (m denotes the area measure on D) and

dτϕdm

(z) = c(1− |z|2)(1−dimEϕ)/2

[1 + o(1)] (|z| 1)

where c = c(ϕ) > 0 and dim refers to the topological dimension. See [2] and [3]. Since Eϕcontains a circle, dimEϕ ≥ 1 and therefore

∫

S2

dσ2

1− |ϕ|2 =∫

D

dτϕ(z)1− |z|2 =∞.

In other words, Cϕ is not a Hilbert-Schmidt operator.

5. Remarks

Note that the a priori boundedness of Cϕ plays an essential role in the argument usedin Section 3, especially in the proof of Lemma 3.2. The situation is different in higherdimensional cases. The densely defined composition operator CΦ of H2(Bk) into H2(Bn)induced by a general holomorphic map Φ : Bn −→ Bk is not necessarily bounded; see forexample [8]. Nevertheless, it is not hard to see that the analogue of 1.(2) still remains true:

||CΦ||2 = ||IτΦ ||2 ∼= ||τΦ ||C

where τΦ = σn[(Φ∗)−1] (* denotes the radial limit) is the pullbcak measure of Φ as before.Now, once the boundedness of CΦ is known, one can easily prove an analogue of Lemma3.2. The rest of the proof of the following theorem remains the same:

5.1. Theorem. Suppose a holomorphic map Φ : Bn −→ Bk induces a bounded composi-tion operator CΦ : H2(Bk) −→ H2(Bn). Then |||CΦ|||2 = |||IτΦ |||2 ∼= |||τΦ |||C .

The following consequence of Theorem 5.1 appears in [8] in case k = n.

5.2. Corollary. Let CΦ be the composition operator induced by a holomorphic map Φ :Bn −→ Bk. Then CΦ is compact if and only if |||τΦ |||C = 0.

Using the above corollary, we give a class of holomorphic maps that induces compactcomposition operators.

5.3. Example. Let Φ : Bn+m −→ Bm be a holomorphic map of the form:

Φ(z, w) = ϕ(z)w (|z|2 + |w|2 < 1, z ∈ Cn, w ∈ Cm)

where ϕ : Bn −→ C is a (possibly unbounded) holomorphic function. Then, by 4.4.(1),

(1)∫

Bm

f dτΦ =∫

Bn

dµ(z)∫

Sm

f(ϕ(z)√

1− |z|2ζ) dσm(ζ)

where f ∈ L1(τΦ) and dµ(z) =(n+m−1

n

)(1 − |z|2)m−1dVn(z). Let EΦ be the set of points

η ∈ Sn+m such that |Φ∗(η)| = 1 as before. Then τΦ(Sm) = σn+m(EΦ). Now, by applying

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(1) to the characteristic function of Sm, we obtain σn+m(EΦ) = µ(E) where E = z ∈Bn : |ϕ(z)|2(1 − |z|2) = 1. Note that E is the zero set of the nontrivial real-analyticfunction |ϕ(z)|2(1 − |z|2) − 1 on Bn and thus µ(E) = 0. It is clear from (1) that τΦ

is rotation-invariant. In summary: τΦ is a rotation-invariant measure on Bm. Thus, byLemma 4.5, |||τΦ |||C = 0. In conclusion: Φ induces a compact composition operatorCΦ : H2(Bm) −→ H2(Bn+m) by Corollary 5.2.

5.4. REMARK. Let α > −1 and define dµα(z) = (1−|z|2)αdV (z) (recall V = Vn denotesthe normalized volume measure on B ). The α-weighted Bergman space A2

α(B) is theclosed subspace of L2(µα) consisting of holomorphic functions. For a measure µ on Bdefine

||µ||α = supµ(Ω(ζ, t))µα(Ω(ζ, t))

(ζ ∈ S, t > 0)

and

|||µ|||α = lim supt0

supζ∈S

µ(Ω(ζ, t))µα(Ω(ζ, t))

.

There is an analogue of the Carleson measure theorem in this context ([4], [6], [17]; also[9]): ||Jµ||2 ∼= ||µ||α where Jµ denotes the densely defined inclusion operator of A2

α(B) intoL2(µ). Thus, if CΦ,α denotes the densely defined composition operator of A2

α(Bk) intoA2α(Bn) induced by a holomorphic map Φ : Bn −→ Bk and if µΦ = µα[Φ−1], then one

easily obtains as before||CΦ,α||2 = ||JµΦ

||2 ∼= ||µΦ ||α.Also, an easy modification of the proof of Proposition 3.4 shows |||Jµ|||2 ∼= |||µ|||α andtherefore one has the following analogue of Theorem 5.1:

|||CΦ,α|||2 = |||JµΦ|||2 ∼= |||µΦ |||α

for bounded operators CΦ,α. In fact the above is much easier to prove because the boundaryfunctions are not involved. Note that CΦ,α is a priori bounded for k = 1. However, CΦ,α

is not bounded for general k; such an example can be found in [9].

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Department of MathematicsKorea Advanced Institute of Science and TechnologyP.O. Box 150, CheongryangSeoul 130-650, KOREA

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