essential norms of composition operators between weighted bergman spaces of the unit disc

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Acta Mathematica Sinica, English Series Apr., 2013, Vol. 29, No. 4, pp. 633–638 Published online: November 8, 2012 DOI: 10.1007/s10114-012-0070-y Http://www.ActaMath.com Acta Mathematica Sinica, English Series © Springer-Verlag Berlin Heidelberg & The Editorial Office of AMS 2013 Essential Norms of Composition Operators between Weighted Bergman Spaces of the Unit Disc Luo LUO Jing CHEN Department of Mathematics, University of Science and Technology of China, Hefei 230026, P. R. China E-mail : [email protected] [email protected] Abstract In this paper, we express the essential norms of composition operators between weighted Bergman spaces of the unit disc in terms of the generalized Nevanlinna counting function. Keywords Essential norm, composition operators, weighted Bergman spaces MR(2010) Subject Classification 32A35, 47B33 1 Introduction Let D be the open unit disk in the complex plane and H(D) denote the space of all holomor- phic functions in D. For α> 1, we define the weighted measure dA α on D by dA α (w)= [log(1/|w|)] α dA(w), where dA is the normalized Lebesgue measure on D. For 0 <p< and α> 1, the weighted Bergman space A p α (D) is defined to be those functions f H(D) satisfying f A p α = D |f (w)| p dA α (w) 1 p < . In this definition, the measure dA α (w) can be replaced by the measure (1 −|w|) α dA(w) (see [1]). The two different measures result in the same space of functions and an equivalent norm, since (1 −|w|) α and [log(1/|w|)] α are comparable for 1 2 ≤|w| < 1, and the singularity of dA α at the origin is integrable. For α = 0, A p 0 (D) is the classical Bergman space. For 1 <p< , A p α (D) is a Banach space. Let ϕ : D D be a holomorphic self-map of D. For a holomorphic function f on D, denote the composition f ϕ by C ϕ f and call C ϕ the composition operator induced by ϕ. Let X and Y be Banach spaces. For a bounded linear operator T : X Y , the essential norm T e,XY is defined to be the distance from T to the set of the compact operators K : X Y , namely, T e,XY = inf {T K : K is compact from X into Y }, where · denotes the usual operator norm. Clearly, T is compact if and only if T e,XY = 0. Thus, the essential norm is closely related to the compactness problem of concrete operators. Received February 4, 2010, accepted March 9, 2012 Supported by National Natural Science Foundation of China (Grant Nos. 11071230 and 11171318) and Natural Science Foundation of Anhui Province (Grant No. 090416233)

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Acta Mathematica Sinica, English Series

Apr., 2013, Vol. 29, No. 4, pp. 633–638

Published online: November 8, 2012

DOI: 10.1007/s10114-012-0070-y

Http://www.ActaMath.com

Acta Mathematica Sinica, English Series© Springer-Verlag Berlin Heidelberg & The Editorial Office of AMS 2013

Essential Norms of Composition Operators between Weighted

Bergman Spaces of the Unit Disc

Luo LUO Jing CHENDepartment of Mathematics, University of Science and Technology of China,

Hefei 230026, P. R. China

E-mail : [email protected] [email protected]

Abstract In this paper, we express the essential norms of composition operators between weighted

Bergman spaces of the unit disc in terms of the generalized Nevanlinna counting function.

Keywords Essential norm, composition operators, weighted Bergman spaces

MR(2010) Subject Classification 32A35, 47B33

1 Introduction

Let D be the open unit disk in the complex plane and H(D) denote the space of all holomor-phic functions in D. For α > −1, we define the weighted measure dAα on D by dAα(w) =[log(1/|w|)]αdA(w), where dA is the normalized Lebesgue measure on D. For 0 < p < ∞and α > −1, the weighted Bergman space Apα(D) is defined to be those functions f ∈ H(D)satisfying

‖f‖Apα

=[∫

D

|f(w)|pdAα(w)] 1

p

<∞.

In this definition, the measure dAα(w) can be replaced by the measure (1−|w|)αdA(w) (see [1]).The two different measures result in the same space of functions and an equivalent norm, since(1 − |w|)α and [log(1/|w|)]α are comparable for 1

2 ≤ |w| < 1, and the singularity of dAα at theorigin is integrable.

For α = 0, Ap0(D) is the classical Bergman space. For 1 < p <∞, Apα(D) is a Banach space.Let ϕ : D → D be a holomorphic self-map of D. For a holomorphic function f on D, denote

the composition f ◦ ϕ by Cϕf and call Cϕ the composition operator induced by ϕ.Let X and Y be Banach spaces. For a bounded linear operator T : X → Y , the essential

norm ‖T‖e,X→Y is defined to be the distance from T to the set of the compact operatorsK : X → Y , namely,

‖T‖e,X→Y = inf{‖T −K‖ : K is compact from X into Y },where ‖ · ‖ denotes the usual operator norm.

Clearly, T is compact if and only if ‖T‖e,X→Y = 0. Thus, the essential norm is closelyrelated to the compactness problem of concrete operators.

Received February 4, 2010, accepted March 9, 2012

Supported by National Natural Science Foundation of China (Grant Nos. 11071230 and 11171318) and Natural

Science Foundation of Anhui Province (Grant No. 090416233)

634 Luo L. and Chen J.

Shapiro [2] expressed the essential norm of the composition operator Cϕ : A2α(D) → A2

α(D)in terms of the generalized Nevanlinna counting function of the inducing map ϕ.

The natural Nevanlinna counting function for ϕ, Nϕ, provides such a measure. It is definedby

Nϕ(w) =∑

z∈ϕ−1{w}log(1/|z|), w ∈ D\{ϕ(0)}.

As usual, z ∈ ϕ−1{w} is repeated according to the multiplicity of the zero of ϕ− w at z.The generalized Nevanlinna counting functions Nϕ,α is defined for γ > 0 by

Nϕ,γ(w) =∑

z∈ϕ−1{w}[log(1/|z|)]γ , w ∈ D\{ϕ(0)}.

Thus Nϕ,1 is the natural Nevanlinna counting function.Smith [3] used these generalized Nevanlinna counting functions to characterize those ϕ that

induce bounded and compact composition operators between weighted Bergman and Hardyspaces.

The main goal of this paper is to compute the essential norm of Cϕ : Apα(D) → Aqβ(D) for1 < p ≤ q <∞ in terms of the generalized Nevanlinna counting function of the inducing map ϕ.

In this paper, we get the following theorem.

Theorem 1.1 Let ϕ be a holomorphic self-map of D, α, β > −1, 1 < p ≤ q < ∞, ifCϕ : Apα(D) → Aqβ(D) is bounded, then, there are constants C1 and C2, such that

C1 lim sup|a|→1−

Nϕ,β+2(a)

[log(1/|a|)] (α+2)qp

≤ ‖Cϕ‖qe,Apα(D)→Aq

β(D)≤ C2 lim sup

|a|→1−

Nϕ,β+2(a)

[log(1/|a|)] (α+2)qp

.

Particularly, we get the corollary.

Corollary 1.2 For α, β > −1, 1 < p ≤ q <∞, then Cϕ : Apα(D) → Aqβ(D) is compact if andonly if

lim sup|a|→1−

Nϕ,β+2(a)

[log(1/|a|)] (α+2)qp

= 0.

Corollary 1.2 was gotten by Smith [3].In the case α = β, p = q = 2, Theorem 1.1 and Corollary 1.2 were gotten by Shapiro [2].Throughout the paper, C denotes a positive constant, whose value may change from one

occurrence to the next one, but is independent of f and ϕ.

2 Proof of Theorem 1.1

Recall that a holomorphic function f in D has the Taylor expansion

f(z) =∞∑k=0

akzk.

For the Taylor expansion of f and any integer n ≥ 1, let

Rnf(z) =∞∑k=n

akzk

and Kn = I − Rn where If = f is the identity operator.The operator Kn has a connection with the following natural question: When does the

partial sums of the Taylor expansion of f converge to f in the norm topology of the function

Essential Norms of Composition Operators 635

space? Zhu [4] have considered the question for various analytic function spaces on the unitdisc. In order to prove our main result, we need some of his results.

Lemma 2.1 Suppose X is a Banach space of holomorphic functions in D with the propertythat the polynomials are dense in X. Then ‖Knf − f‖X → 0 as n → ∞ if and only ifsup{‖Kn‖ : n ≥ 1} <∞.

Lemma 2.2 If 1 < p <∞, then ‖Knf−f‖Apα→ 0 as n→ ∞ for each f ∈ Apα(D). Moreover,

sup{‖Rn‖ : n ≥ 1} <∞ and sup{‖Kn‖ : n ≥ 1} <∞.

Lemmas 2.1 and 2.2 are Proposition 1 of [4].To prove Theorem 1.1, we also need the following lemmas.

Lemma 2.3 For α > −1, 0 < p <∞, f ∈ H(D) and ϕ is a holomorphic self-map of D, then

‖f ◦ ϕ‖pAp

α≈ |f(ϕ(0))|p +

∫D

|f(w)|p−2|f ′(w)|2Nϕ,α+2(w)dA(w),

where the symbol “≈” means that the left-hand side is bounded above and below by constantmultiples of the right-hand side, where the constants are positive and independent of f .

Lemma 2.3 is Proposition 2.4 in [3].

Lemma 2.4 Let ψ be a holomorphic self-map of D and γ > 0. If ψ(0) = 0 and 0 < r < |ψ(0)|,then

Nψ,γ(0) ≤ 1r2

∫rD

Nψ,γ(w)dA(w).

Lemma 2.4 is Lemma 4.1 in [3].

Lemma 2.5 Let ψ be a holomorphic self-map of D and γ > 0. Let a ∈ D and

σa(w) =a− w

1 − aw

be the Mobius self-map of D that interchanges 0 and a. Then

(Nψ,γ) ◦ σa = Nσa◦ψ,γ .

Lemma 2.5 is Lemma 4.2 in [3].

Lemma 2.6 For α > −1, 0 < p <∞, f ∈ Apα(D) and w ∈ D, then

|f(w)| ≤ C‖f‖Apα

(1 − |w|)α+2p

,

where C is independent of f .

Lemma 2.6 is Lemma 2.5 in [3].

Lemma 2.7 For 0 < p <∞, s ≥ 0, 2 + s ≥ p, f ∈ Apα(D) and w ∈ D, then

|f ′(w)|p ≤ C

∫D

|f(z)|p(1 − |z|2)s|1 − wz|s+p+2

dA(w),

where C is independent of f .

Lemma 2.7 is the special case of Lemma 2 in [5].

Proof of Theorem 1.1 At first, we prove

‖Cϕ‖qe,Apα(D)→Aq

β(D)≥ C1 lim sup

|a|→1−

Nϕ,β+2(a)

[log(1/|a|)] (α+2)qp

.

636 Luo L. and Chen J.

For a ∈ D, let

ka(z) ={

1 − |a|2(1 − az)2

} 2+αp

,

we know ‖ka‖Apα

= 1 and, as |a| → 1−, ka → 0 uniformly on compact subset of D.For the moment fixing a compact operator K : Apα(D) → Aqβ(D). Since the family {ka} is

bounded in Apα(D), and ka → 0 uniformly on compact subsets of D as |a| → 1−, we also have‖Kka‖Aq

β→ 0, as |a| → 1−. Thus

‖Cϕ −K‖ ≥ lim sup|a|→1−

‖(Cϕ −K)ka‖Aqβ≥ lim sup

|a|→1−(‖Cϕka‖Aq

β− ‖Kka‖Aq

β)

= lim sup|a|→1−

‖Cϕka‖Aqβ.

Upon taking the infimum of both sides of this inequality over all compact operators K :Apα(D) → Aqβ(D), we obtain

‖Cϕ‖e,Apα(D)→Aq

β(D) ≥ lim sup|a|→1−

‖Cϕka‖Aqβ. (2.1)

By Lemma 2.3,

‖Cϕka‖qAqβ≈ |ka(ϕ(0))|q +

∫D

|ka(w)|q−2|k′a(w)|2Nϕ,β+2(w)dA(w).

So, there is a constant C such that

‖Cϕka‖qAqβ≥ C

∫D

|ka(w)|q−2|k′a(w)|2Nϕ,β+2(w)dA(w)

= C4(2 + α)2

p2|a|2(1 − |a|2) (2+α)q

p

∫D

Nϕ,β+2(w)

|1 − aw|2+ 2(2+α)qp

dA(w)

= C4(2 + α)2

p2|a|2(1 − |a|2) (2+α)q

p −2

∫D

Nϕ,β+2(w)

|1 − aw| 2(2+α)qp −2

|σ′a(w)|2dA(w)

= C4(2 + α)2

p2|a|2(1 − |a|2) (2+α)q

p −2

∫D

Nϕ,β+2(σa(z))

|1 − aσa(z)|2(2+α)q

p −2dA(z),

where σa = σ−1a is the Mobius self-map of D as in Lemma 2.5, and the change of variable

z = σa(w) was made. Now,1

|1 − aσa(z)| =|1 − az|1 − |a|2 ≥ 1

2(1 − |a|2) , as |z| ≤ 12,

so

‖Cϕka‖qAqβ≥ C|a|2

(1 − |a|2) (2+α)qp

∫12D

Nϕ,β+2(σa(z))dA(z).

Since σa ◦ ϕ(0) > 12 if |a| is sufficiently close to 1, we can apply Lemmas 2.4 and 2.5, then∫

12D

Nϕ,β+2(σa(z))dA(z) =∫

12D

Nσa◦ϕ,β+2(z)dA(z) ≥ 4Nσa◦ϕ,β+2(0) = 4Nϕ,β+2(a).

Therefore,

‖Cϕka‖qAqβ≥ C|a|2Nϕ,β+2(a)

(1 − |a|2) (2+α)qp

.

Since log(1/|a|) is comparable to (1− |a|2) if |a| is sufficiently close to 1, thus, by (2.1), we get

‖Cϕ‖qe,Apα(D)→Aq

β(D)≥ C1 lim sup

|a|→1−

Nϕ,β+2(a)

[log(1/|a|)] (2+α)qp

.

Essential Norms of Composition Operators 637

Now, we turn to prove

‖Cϕ‖qe,Apα(D)→Aq

β(D)≤ C2 lim sup

|a|→1−

Nϕ,β+2(a)

[log(1/|a|)] (2+α)qp

.

Since for each n, Kn is compact, so CϕKn is compact and for each n,

‖Cϕ‖e,Apα(D)→Aq

β(D) = ‖CϕRn + CϕKn‖e,Apα(D)→Aq

β(D) ≤ ‖CϕRn‖. (2.2)

Let U denote the closed unit ball in Apα(D), for f(z) ∈ U , by Lemma 2.3,

‖CϕRnf‖qAqβ≤ C

(|Rnf(ϕ(0))|q +

∫D

|Rnf(w)|q−2|(Rnf)′(w)|2Nϕ,β+2(w)dA(w)). (2.3)

For a fixed constant r0, 12 < r0 < 1, we have∫

D

|Rnf(w)|q−2|(Rnf)′(w)|2Nϕ,β+2(w)dA(w)

=∫r0D

|Rnf(w)|q−2|(Rnf)′(w)|2Nϕ,β+2(w)dA(w)

+∫D\r0D

|Rnf(w)|q−2|(Rnf)′(w)|2Nϕ,β+2(w)dA(w). (2.4)

Let M = sup|w|>r0Nϕ,β+2(w)

[log(1/|w|)](2+α)q

p

, by Lemma 2.6,

|Rnf(w)|q−p ≤ C‖Rnf‖q−pApα

(1 − |w|) (2+α)(q−p)p

.

Then ∫D\r0D

|Rnf(w)|q−2|(Rnf)′(w)|2Nϕ,β+2(w)dA(w)

≤ CM‖Rnf‖q−pApα

·∫D\r0D

|Rnf(w)|p−2|(Rnf)′(w)|2 [log(1/|w|)] (2+α)(q−p)p [log(1/|w|)](2+α)

(1 − |w|) (2+α)(q−p)p

dA(w).

Since log(1/|w|) ≤ 2(1 − |w|) as |w| ≥ 12 , then∫

D\r0D|Rnf(w)|q−2|(Rnf)′(w)|2Nϕ,β+2(w)dA(w)

≤ CM‖Rnf‖q−pApα

∫D\r0D

|Rnf(w)|p−2|(Rnf)′(w)|2[log(1/|w|)](2+α)dA(w).

For ϕ(z) = z, then Nϕ,α+2(w) = [log(1/|w|)](2+α), by Lemma 2.3,∫D\r0D

|Rnf(w)|p−2|(Rnf)′(w)|2[log(1/|w|)](2+α)dA(w) ≤ C‖Rnf‖pApα.

By Lemma 2.2 and f(z) ∈ U , we have∫D\r0D

|Rnf(w)|q−2|(Rnf)′(w)|2Nϕ,β+2(w)dA(w) ≤ CM‖Rnf‖qApα≤ CM. (2.5)

Using Lemma 2.7, for s = α+ p,

|(Rnf)′(w)(z)|p ≤ C

∫D

|Rnf(z)|p(1 − |z|2)α+p

|1 − wz|α+p+p+2dA(w),

638 Luo L. and Chen J.

therefore,

|(Rnf)′(w)(z)|p ≤ C

∫D

|Rnf(z)|p(1 − |z|2)α|1 − wz|α+p+2

dA(w).

By the definition of the weighted Bergman space Apα(D), we get

|(Rnf)′(w)(z)| ≤ C‖Rnf‖Apα

(1 − |w|)α+p+2p

.

By Lemma 2.6,

|Rnf(w)|q−2|(Rnf)′(w)|2 ≤ C‖Rnf‖qApα

(1 − |w|) (α+2)(q−1)+pp

.

By Lemma 2.2, ‖Rnf‖Apα→ 0, as n→ ∞, so, as n→ ∞,

|Rnf(w)|q−2|(Rnf)′(w)|2 → 0 uniformly on r0D and |Rnf(ϕ(0))| → 0. (2.6)

By Lemma 2.3, for f(z) = z and p = 2, we get

‖ϕ‖2A2

β= |ϕ(0))|2 + 2

∫D

Nϕ,β+2(w)dA(w).

So, by Lemma 2.6, we get∫r0D

Nϕ,β+2(w)dA(w) ≤ C, where C is independent of ϕ. (2.7)

Combining (2.2)–(2.7) and letting n→ ∞, we get

‖Cϕ‖qe,Apα(D)→Aq

β(D)≤ C sup

|w|>r0

Nϕ,β+2(w)

[log(1/|w|)] (2+α)qp

.

Let r0 → 1−. Then

‖Cϕ‖qe,Apα(D)→Aq

β(D)≤ C2 lim sup

|a|→1−

Nϕ,β+2(a)

[log(1/|a|)] (2+α)qp

.

The proof is end. �

Acknowledgements The authors would like to thank the referee for the careful reading ofthe first version of this paper and for the several suggestions made for improvement.

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