N-differentiation composition operators from weighted ...

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 16, NO.3, 2014, COPYRIGHT 2014 EUDOXUS PRESS, LLC

N-differentiation composition operators from weighted Banach spaces of holomorphic function to weighted Bloch spaces Cui Chen Department of Mathematics, Tianjin University,Tianjin 300072,P.R. China, chencui [email protected] Hong-Gang Zeng∗ Department of Mathematics, Tianjin University,Tianjin 300072,P.R. China, [email protected] Ze-Hua Zhou Department of Mathematics, Tianjin University,Tianjin 300072,P.R. China, [email protected] 1

Abstract In this paper, we characterize nth differentiation composition operators from weighted Banach space of holomorphic function to weighted Bloch space, and give some necessary and sufficient conditions for the boundedness and compactness of the operators.

1

Introduction

Let H(D) and S(D) denote the class of analytic functions and analytic self-maps on the unit disk D of the complex plane of C, respectively. Let v and w be strictly positive continuous and bounded functions (weight) on D. Weighted Banach spaces of holomorphic functions is defined by Hv∞ = {f ∈ H(D) : kf kv := sup v(z)|f (z)| < ∞}, z∈D

endowed with the weighted sup-norm k.kv . An f ∈ H(D) belongs to weighted Bloch spaces Bw if bw (f ) = sup w(z)|f 0 (z)| < ∞. z∈D

The quantity bw (f ) defines a seminorm on Bw , while a natural norm is given by kf kBw = |f (0)| + bw (f ). This norm makes Bw into a Banach space. By Bw,0 we denote the little weighted Bloch space, the subspace of Bw , consisting of all f ∈ Bw such that lim w(z)|f 0 (z)| = 0. |z|→1

Each φ in S(D) induces through composition a linear composition operator Cφ : H(D) → H(D), f 7→ f ◦ φ. And n-differentiating composition operator is a linear operator defined by 1 The authors were supported in part by the National Natural Science Foundation of China (Grant Nos. 10971153, 11126164, 11201331) ∗ Corresponding author. 2010 Mathematics Subject Classification. Primary: 47B38; Secondary: 30H30, 30H05, 47B33, 47G10. Key words and phrases.n-differentiation composition operator, weighted Banach spaces, weighted Bloch space, compact, difference.

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Chen, Zeng and Zhou: N-differentiation composition operators

Dφn : H(D) → H(D), f 7→ f (n) (φ). We are interested in Dφn acting from weighted Banach spaces of holomorphic functions to weighted Bloch spaces. In the setting of weighted spaces the so-called associated weight plays an important role. For a weight v its associated weight ve is defined follows: ve(z) =

1 1 , = sup{|f (z)| : f ∈ Hv∞ , kf kv ≤ 1} kδz kHv∞

where δz denotes the point evaluation of z. By [1] the associated weight ve is continuous, 1 ve ≥ v > 0 and for every z ∈ D we can find fz ∈ Hv∞ with kfz kv ≤ 1 such that |fz (z)| = ve(z) . We say that a weight v is radial if v(z) = v(|z|) for every z ∈ D. A positive continuous function v is called normal if there exist δ ∈ [0, 1) and s, t(0 < s < t) such that for every z ∈ D with |z| ∈ [δ, 1), v(|z|) is decreasing on [δ, 1) and (1 − |z|)s

v(|z|) = 0; |z|→1 (1 − |z|)s

v(|z|) is increasing on [δ, 1) and (1 − |z|)t

v(|z|) = ∞. |z|→1 (1 − |z|)t

lim

lim

A radial, non-increasing weight is called typical if lim v(z) = 0. When studying the struc|z|→1

ture and isomorphism classes of the space Hv∞ (see [6, 7]), Lusky introduced the following condition (L1) (renamed after the author) for radial weights: v(1 − 2−n−1 ) > 0, n∈N 1 − 2−n

(L1) inf

which will play a great role in this article. Moreover, radial weights with (L1) (for example, see [2]) are essential, that is, we can find a constant k > 0 such that v(z) ≤ ve(z) ≤ kv(z) for every z ∈ D. a−z Now, let ϕa (z) = 1−¯ obius transformation that interchanges a and 0. az , z ∈ D, be the M¨ We will use the fact that derivative of ϕa is given by

ϕ0a (z) = −

1 − |a|2 for every z ∈ D. (1 − a ¯z)2

Our aim in this note is to characterize boundedness and compactness of operator Dnφ from weighted Banach spaces of holomorphic functions to weighted Bloch spaces in terms of the involved weights as well as the inducing map. For n = 0 and n = 1, as corollaries we get a characterization of boundedness and compactness of Cφ and Cφ D that act from weighted Banach spaces of holomorphic functions to weighted Bloch spaces. Throughout this paper, we will use the symbol C to denote a finite positive number, and it may differ from one occurrence to the other.

2

Background and Some Lemmas

Now let us state a couple of lemmas, which are used in the proof of the main results in the next sections. The first lemma is taken from [9].

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Chen, Zeng and Zhou: N-differentiation composition operators

Lemma 1. Let v be a radial weight satisfying condition (L1). There is a constant C > 0 (depending only on the weight v) such that for all f ∈ Hv∞ , |f (n) (z)| ≤ C

kf kv , v(z)(1 − |z|2 )n

(1)

for every z ∈ D and n ∈ N. Proof. We will prove the theorem by mathematical induction. For n = 1, see Lemma 2 in [9]. If (1) is true for n − 1. Then for n, let u(z) = v(z)(1 − |z|2 )n−1 , since |f (n−1) (z)| ≤ C

kf kv , v(z)(1 − |z|2 )n−1

then f (n−1) ∈ Hu∞ . For f (n−1) using the result of n = 1 the lemma is proved. The following result is well-known (see, e.g. [3, 8]) Lemma 2. Suppose that w is a normal weight and v is a radial weight satisfying (L1). Then the operator Dφn : Hv∞ → Bw (or Bw,0 ) is compact if and only if whenever {fm } is a bounded sequence in Hv∞ with fm → 0 uniformly on compact subsets of D, and then kDφn fm kBw → 0. The following lemma can be proved similarly to Lemma 1 in [4] (see, also [5]). It will be useful to give a criterion for compactness in Bw,0 . Lemma 3. Assume w is normal. A closed set K in Bw,0 is compact if and only if it is bounded and satisfies lim sup w(z)|f 0 (z)| = 0.

(2)

|z|→1 f ∈K

3

The Boundedness of Dφn : Hv∞ → Bw (or Bw,0 )

In this section we formulate and prove results regarding the boundedness of the operator Dφn : Hv∞ → Bw (or Bw,0 ). Theorem 1. Suppose that w be arbitrary weight, v be a radial weight satisfying condition (L1), then Dφn : Hv∞ → Bw is bounded if and only if sup z∈D

w(z)|φ0 (z)| < ∞, v(φ(z))(1 − |φ(z)|2 )n+1

(3)

Proof. First, we assume that the operator Dφn : Hv∞ → Bw is bounded. Fix a point a ∈ D, and consider the function fa (z) = ϕn+1 (z)ga (z) for every z ∈ D, a where ga is a function in the unit ball of Hv∞ such that ga (a) =

1 v e(a) .

Then

kfa kv = sup v(z)|fa (z)| ≤ sup v(z)|ga (z)| ≤ 1. z∈D

z∈D

It is easy to check that (ϕn+1 )(k) (a) = 0, a

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Chen, Zeng and Zhou: N-differentiation composition operators

(ϕn+1 )(n+1) (a) = a So fa(n+1) (a) =

n+1 X

(−1)n+1 (n + 1)! . (1 − |a|2 )n+1

k Cn+1 (ϕn+1 )(k) (a)ga(n+1−k) (a) = a

k=0

(−1)n+1 (n + 1)! . (1 − |a|2 )n+1 ve(a)

Then by the boundedness of Dφn : Hv∞ → Bw , we have (n+1)

∞ > kDφn fφ(a) kBw ≥ sup w(z)|fφ(a) (φ(z))φ0 (z)| z∈D

(n+1)

≥ w(a)|fφ(a) (φ(a))φ0 (a)| =

(n + 1)!w(a)|φ0 (a)| . (1 − |φ(a)|2 )n+1 ve(φ(a))

Since v has (L1), the weights v and ve are equivalent then ve can be replaced by v, and combine with the arbitrariness of a ∈ D, we obtain (3). Conversely, an application of Lemma 1 yields w(z)|f (n+1) (φ(z))φ0 (z)| ≤ C

w(z)|φ0 (z)| kf kv , v(φ(z))(1 − |φ(z)|2 )n+1

(4)

and |f (n) (φ(0))| ≤ C

kf kv . v(φ(0))(1 − |φ(0)|2 )n

Combine with this and taking the supremum in (4) over D, then employing condition (3), we see that Dφn : Hv∞ → Bw must be bounded. By the similar proof of Theorem 1 we see that the following result is true. Theorem 2. Suppose that w be arbitrary weight, v be a radial weight satisfying condition (L1), then Dφn : Hv∞ → Bw,0 is bounded if and only if w(z)|φ0 (z)| = 0. |z|→1 v(φ(z))(1 − |φ(z)|2 )n+1 lim

(5)

Especially, for n = 0, we obtain necessary and sufficient conditions for the boundedness of the operators Cφ : Hv∞ → Bw (or Bw,0 ). Corollary 1. Suppose that w be arbitrary weight, v be a radial weight satisfying condition (L1), then the following statements hold: (i) Cφ : Hv∞ → Bw is bounded if and only if sup z∈D

w(z)|φ0 (z)| < ∞. v(φ(z))(1 − |φ(z)|2 )

(ii) Cφ : Hv∞ → Bw,0 is bounded if and only if w(z)|φ0 (z)| = 0. |z|→1 v(φ(z))(1 − |φ(z)|2 ) lim

For n = 1, Dφn is the operator Cφ D, then we have the following corollary .

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Chen, Zeng and Zhou: N-differentiation composition operators

Corollary 2. Suppose that w be arbitrary weight, v be a radial weight satisfying condition (L1), then the following statements hold: (i) Cφ D : Hv∞ → Bw is bounded if and only if sup z∈D

(ii) Cφ D :

Hv∞

w(z)|φ0 (z)| < ∞. v(φ(z))(1 − |φ(z)|2 )2

→ Bw,0 is bounded if and only if w(z)|φ0 (z)| = 0. |z|→1 v(φ(z))(1 − |φ(z)|2 )2 lim

4

The Compactness of Dφn : Hv∞ → Bw (or Bw,0 )

In this section, we turn our attention to the question of compactness. Theorem 3. Suppose that w be arbitrary weight, v be a radial weight satisfying condition (L1). Then Dφn : Hv∞ → Bw is compact if and only if lim

sup

r→1 |φ(z)|>r

w(z)|φ0 (z)| = 0. v(φ(z))(1 − |φ(z)|2 )n+1

(6)

Proof. First, we assume that the operator Dφn : Hv∞ → Bw is compact. Let {zm }m ⊂ D be a sequence with |φ(zm )| → 1 such that lim

sup

r→1 |φ(z)|>r

w(zm )|φ0 (zm )| w(z)|φ0 (z)| = lim . m→∞ v(φ(zm ))(1 − |φ(zm )|2 )n+1 v(φ(z))(1 − |φ(z)|2 )n+1

By passing to a subsequence and still denoted by {zm }m , we assume that there is N ∈ N, such that |φ(zm )|m ≥ 12 for every m ≥ N . For every m ∈ N, we consider functions fm (z) = z m ϕn+1 φ(zm ) (z)gφ(zm ) (z) for every z ∈ D, 1 where gφ(zm ) is a function in the unit ball of Hv∞ such that |gφ(zm ) (φ(zm ))| = ve(φ(z . m )) ∞ Again since v has (L1), ve may be replaced by v. Obviously, {fm }m ⊂ Hv is a bounded sequence that tends to zero uniformly on the compact subsets of D. Hence by Lemma 2, we have that kDφn fm kBw → 0. Moreover, (k) (z m ϕn+1 (φ(zm )) = 0, φ(zm ) ) (n+1) (z m ϕn+1 (φ(zm )) = φ(zm ) )

Since (n+1) fm (φ(zm ))

=

n+1 X

k = 0, 1, ..., n;

(−1)n+1 (n + 1)!φm (zm ) . (1 − |φ(zm )|2 )n+1 (n+1−k)

k (k) Cn+1 (z m ϕn+1 gφz φ(zm ) )

m

(φ(zm )).

k=0 (n+1)

Therefore |fm

(φ(zm ))| = 0 ← = ≥

(n+1)!|φ(zm )|m v e(φ(zm ))(1−|φ(zm )|2 )n+1 ,

and for m ≥ N

(n+1) kDφn fm kBw ≥ w(zm )|fm (φ(zm ))φ0 (zm )|

(n + 1)!w(zm )|φ0 (zm )||φ(zm )|m ve(φ(zm ))(1 − |φ(zm )|2 )n+1 1 w(zm )|φ0 (zm )| , 2 v(φ(zm ))(1 − |φ(zm )|2 )n+1

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Chen, Zeng and Zhou: N-differentiation composition operators

and the claim follows. Conversely, suppose that (6) holds. Let {fm }m ⊂ Hv∞ be a bounded sequence which converges to zero uniformly on the compact subsets of D, we may assume that kfm kv ≤ 1 for every m ∈ N. By Lemma 2 we have to show that kDφn fm kBw → 0 if m → ∞. Let us fix ε > 0. By hypothesis there is 0 < r < 1 such that ε w(z)|φ0 (z)| < if |φ(z)| > r, v(φ(z))(1 − |φ(z)|2 )n+1 2C where C is the constant given in Lemma 1. Thus, if |φ(z)| > r, by Lemma 1, (n+1) w(z)|φ0 (z)||fm (φ(z))| ≤ C

w(z)|φ0 (z)| ε kfm kv < . v(φ(z))(1 − |φ(z)|2 )n+1 4

(7)

Now, we can find M > 0 such that sup w(z)|φ0 (z)| ≤ M .

(8)

|φ(z)|≤r

Moreover, since {fm }m converges to 0 uniformly on compact subsets of D as m → ∞. (n+1) }m also converges to 0 uniformly on compact Cauchy’s integral formula gives that {fm subsets of D as m → ∞. So there is N1 ∈ N such that ε for every m ≥ N1 . 4M

(n+1) sup |fm (φ(z))| ≤

|φ(z)|≤r

(9)

(n)

Also, {fm (φ(0))}m converges to 0 as m → ∞, then there exists N2 > 0 such that (n) |fm (φ(0))| < 2ε for every m > N2 . Finally, together with (7) (8) and (9) we can conclude that kDφn fm kBw

(n) (n+1) = |fm (φ(0))| + sup w(z)|φ0 (z)||fm (φ(z))| z∈D

≤

(n) |fm (φ(0))|

(n+1) + sup w(z)|φ0 (z)| sup |fm (φ(z))| |φ(z)|≤r 0

+ sup w(z)|φ |φ(z)|>r


0 there exists a r ∈ (0, 1) such that w(z)|φ0 (z)| < ε if r < |φ(z)| < 1. v(φ(z))(1 − |φ(z)|2 )n+1

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Chen, Zeng and Zhou: N-differentiation composition operators

n+1

z On the other hand, since h(z) = (n+1)! ∈ Hv∞ , from the compactness of Dφn : Hv∞ → Bw,0 , it follows that φ ∈ Bw,0 . Then there exists a ρ ∈ (r, 1) such that

w(z)|φ0 (z)| < ε inf v(t)(1 − |t|2 )n+1 if ρ < |z| < 1,

(11)

t∈[0,r]

Therefore, when ρ < |z| < 1 and r < |φ(z)| < 1, we have that w(z)|φ0 (z)| < ε. v(φ(z))(1 − |φ(z)|2 )n+1

(12)

If ρ < |z| < 1 and |φ(z)| ≤ r, combine with (11), we have that w(z)|φ0 (z)| w(z)|φ0 (z)| ≤ < ε. 2 n+1 v(φ(z))(1 − |φ(z)| ) inf v(t)(1 − |t|2 )n+1

(13)

t∈[0,r]

Inequalities (12) and (13) imply (10) holds. Conversely, assume that (10) holds. Then (3) holds, which along with (4) implies that the set Dφn ({f ∈ Hv∞ : kf kv ≤ 1}) is bounded in Bw,0 . By Lemma 3 we see that Dφn : Hv∞ → Bw,0 is compact if and only if sup w(z)|f (n+1) (φ(z))φ0 (z)| = 0..

lim

|z|→1 kf kv ≤1

(14)

Taking the supremum in (4) over the unit ball of Hv∞ , then letting |z| → 1, we obtain (14), from which the compactness of Dφn : Hv∞ → Bw,0 follows. Noticing the results of Theorem 2 and Theorem 4, we conclude that the boundedness and compactness of the operator Dφn : Hv∞ → Bw,0 is equivalent. Similarly, for n = 0, we obtain necessary and sufficient conditions for the compactness of the operators Cφ : Hv∞ → Bw (or Bw,0 ). Corollary 3. Suppose that w be a normal weights, v be a radial weight satisfying condition (L1). Then the following statements hold: (i) Cφ : Hv∞ → Bw is compact if and only if lim

sup

r→1 |φ(z)|>r

w(z)|φ0 (z)| = 0. v(φ(z))(1 − |φ(z)|2 )

(ii) Cφ : Hv∞ → Bw,0 is compact if and only if w(z)|φ0 (z)| = 0. |z|→1 v(φ(z))(1 − |φ(z)|2 ) lim

And for n = 1, Dφn is the operator Cφ D. Corollary 4. Suppose that w be a normal weights, v be a radial weight satisfying condition (L1). Then the following statements hold: (i) Cφ D : Hv∞ → Bw is compact if and only if lim

sup

r→1 |φ(z)|>r

w(z)|φ0 (z)| = 0. v(φ(z))(1 − |φ(z)|2 )2

(ii) Cφ D : Hv∞ → Bw,0 is compact if and only if w(z)|φ0 (z)| = 0. |z|→1 v(φ(z))(1 − |φ(z)|2 )2 lim

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Chen, Zeng and Zhou: N-differentiation composition operators

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