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