Essential norm of generalized weighted composition operators on Bloch-type spaces

Applied Mathematics and Computation 274 (2016) 133–142

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Essential norm of generalized weighted composition operators on Bloch-type spaces Xiangling Zhu a,b,∗ a b

Faculty of Information Technology, Macau University of Science and Technology, Avenida Wai Long, Taipa, Macau Department of Mathematics, Jiaying University, 514015 Meizhou, Guangdong, China

a r t i c l e

i n f o

a b s t r a c t

MSC: 30H30 47B38 Keywords: Bloch-type space Essential norm Generalized weighted composition operator

In this paper, we give some estimates of the essential norm for generalized weighted composition operators on Bloch-type spaces. Moreover, we give a new characterization for the boundedness and compactness of the generalized weighted composition operator on Blochtype spaces. © 2015 Elsevier Inc. All rights reserved.

1. Introduction Let D be the open unit disk in the complex plane C and H (D) be the space of analytic functions on D. Let α ∈ (0, ∞). An f ∈ H (D) is said to belong to the Bloch-type space (or the α -Bloch space), denoted by B α , if α

 f α = sup (1 − |z|2 ) | f  (z)| < ∞. z∈D

B α is a Banach space under the norm  f Bα = | f (0)| +  f α . When α = 1, B 1 = B is the classical Bloch space. An f ∈ B α is said α to belong to the little Bloch type space B0α (or the little α -Bloch space) if lim|z|→1 | f  (z)|(1 − |z|2 ) = 0. See [35] for the theory of Bloch-type spaces. Let ϕ be a nonconstant analytic self-map of D and u ∈ H (D). The weighted composition operator, denoted by uCϕ , is defined as follows:

uCϕ f = u(z) · f (ϕ(z)), f ∈ H (D). When u = 1, we get the composition operator, denoted by Cϕ . When ϕ(z) = z, we get the multiplication operator, denoted by Mu . Let n be a nonnegative integer. A linear operator, denoted by Dnϕ ,u , is defined as follows (see, e.g., [36]):

(Dnϕ ,u f )(z) = u(z) · f (n) (ϕ(z)), f ∈ H (D), z ∈ D. This operator is called the generalized weighted composition operator and was introduced by the author of this paper, motivated by the previous study of products of composition and differentiation operators (see, e.g., [2,7]). When n = 0, Dnϕ ,u = uCϕ . When n = 1 and u(z) = ϕ  (z), then Dnϕ ,u = DCϕ , which was studied in [2,7,9,11,13,19,21,32]. When u(z) = 1, then Dnϕ ,u = Cϕ Dn , which ∗

Corresponding author at: Department of Mathematics, Jiaying University, 514015 Meizhou, Guangdong, China. Tel.: +86 13035778815. E-mail address: [email protected]

http://dx.doi.org/10.1016/j.amc.2015.10.061 0096-3003/© 2015 Elsevier Inc. All rights reserved.

134

X. Zhu / Applied Mathematics and Computation 274 (2016) 133–142

was studied, for example, in [2,19,30]. See, for example, [5,23–25,33,36–39] for the study of the generalized weighted composition operator on various function spaces. Recently there has been a huge interest in the study of concrete product-type operators on various domains in the complex plane C or the n-dimensional complex space Cn . For some other product-type operators containing composition operators, see, e.g., [12,20,22,26–28] and the reference therein. Various properties of composition operator, as well as weighted composition operators on Bloch-type spaces were studied, for example, in [1,2,8,10,11,14–16,18,21,25,29–32,34,39]. Tjani in [29] proved that Cϕ : B → B is compact if and only if

lim

|a|→1

      1 − |a|2  a−z    = lim Cϕ  = 0.  Cϕ 1 − az ¯ ¯ 1 − az |a|→1

B

B

Wulan, Zheng and Zhu obtained a new characterization for the compactness of the composition operator Cϕ : B → B in [31], i.e., they proved that Cϕ : B → B is compact if and only if lim j→∞ ϕ j B = 0. In [34], Zhao extended the result in [31] to Bloch-type spaces. In particular, he obtained the exact value for the essential norm of Cϕ : B α → B β as follows:

Cϕ e,Bα →Bβ =

 e α lim sup nα −1 ϕ n β . 2α n→∞

Recall that the essential norm of a bounded linear operator T: X → Y is its distance to the set of compact operators K mapping X into Y, that is,

T e,X→Y = inf{T − K X→Y : K is compact}, where X, Y are Banach spaces and  · X → Y is the operator norm. Ohno, Stroethoff and Zhao studied the boundedness and compactness of the operator uCϕ : B α → B β in [18]. The essential norm of the operator uCϕ : B α → B β was given in [14]. Manhas and Zhao obtained some new estimates for the essential norm of uCϕ : B α → B β in [16]. In particular, when α > 1, they obtained the following result. Theorem A. Suppose α > 1 and 0 < β < ∞ and suppose that uCϕ : B α → B β is bounded. Then

  uCϕ e,Bα →Bβ ≈ max lim sup jα−1 Iu (ϕ j )Bβ , lim sup jα−1 Ju (ϕ j )Bβ , j→∞

where

Iu f (z) =



z 0

f  (ζ )u(ζ )dζ , Ju f (z) =

j→∞



z 0

f (ζ )u (ζ )dζ .

In [30], Wu and Wulan proved that Cϕ Dn : B → B is compact if and only if lim|a|→1 Cϕ Dn

[39], we consider the case of the operator Dnϕ ,u : B α → B β and obtained the following result.



a−z ¯ 1−az



B = 0. Motivated by this, in

Theorem B. Let n be a positive integer, 0 < α , β < ∞, u ∈ H (D) and ϕ be an analytic self-map of D such that Dnϕ ,u : B α → B β is bounded. Then the following statements are equivalent. (a) Dnϕ ,u : B α → B β is compact. (b)

   1 − |a|2   n lim Dϕ ,u  = 0 and |a|→1 (1 − az)α Bβ

  2  (1 − |a|2 )   n lim Dϕ ,u  = 0. |a|→1 (1 − az)α+1 Bβ

(c) β

(1 − |z|2 ) |u(z)||ϕ  (z)| = 0 and |ϕ(z)|→1 (1 − |ϕ(z)|2 )n+α lim

β

(1 − |z|2 ) |u (z)| = 0. |ϕ(z)|→1 (1 − |ϕ(z)|2 )n+α −1 lim

Motivated by [6], the purpose of this paper is to give some estimates of the essential norm for the operator Dnϕ ,u : B α → B β .

Moreover, we give a new characterization for the boundedness, compactness and essential norm of the operator Dnϕ ,u : B α → B β . Throughout this paper, we say that P ࣠ Q if there exists a constant C such that P ≤ CQ. The symbol P ≈ Q means that P ࣠ Q ࣠ P. 2. Essential norm of Dnϕ,u : Bα → Bβ In this section, we give two estimates of the essential norm for the operator Dnϕ ,u : B α → B β . Theorem 2.1. Let n be a positive integer, 0 < α , β < ∞, u ∈ H (D) and ϕ be an analytic self-map of D such that Dnϕ ,u : B α → B β is bounded. Then

Dnϕ ,u e,Bα →Bβ ≈ max{A, B } ≈ max{E, F },

X. Zhu / Applied Mathematics and Computation 274 (2016) 133–142

where

135

     2   n  n 1 − |a|2  (1 − |a|2 )    ,   A := lim sup Dϕ ,u , B := lim sup Dϕ ,u (1 − az)α+1  β (1 − az)α  β |a|→1 |a|→1 B

B

β

E := lim sup |ϕ(z)|→1

β

(1 − |z|2 ) |u(z)||ϕ  (z)| (1 − |z|2 ) |u (z)| , F := lim sup . n+α n+α −1 2 (1 − |ϕ(z)| ) |ϕ(z)|→1 (1 − |ϕ(z)|2 )

Proof. First we prove that max{A, B} ≤ Dnϕ ,u e,Bα →Bβ . Let a ∈ D. Define

fa (z) =

2

1 − |a|2

ga (z) =

α,

(1 − az)

(1 − |a|2 ) , z ∈ D. (1 − az)α+1

It is easy to check that fa , ga ∈ B0α and  fa Bα  1, ga Bα  1 for all a ∈ D and fa , ga converges to 0 weakly in B α as |a| → 1. This follows since a bounded sequence contained in B0α which converges uniformly to 0 on compact subsets of D converges weakly to 0 in B α (see [14]). Thus, for any compact operator K : B α → B β , we have

lim

|a|→1

K fa Bβ = 0,

lim

|a|→1

Kga Bβ = 0.

Hence

Dnϕ ,u − K Bα →Bβ  lim sup (Dnϕ ,u − K ) fa Bβ |a|→1

≥ lim sup Dnϕ ,u fa Bβ − lim sup K fa Bβ = A, |a|→1

|a|→1

and

Dnϕ ,u − K Bα →Bβ  lim sup (Dnϕ ,u − K )ga Bβ |a|→1

≥ lim sup Dnϕ ,u ga Bβ − lim sup Kga Bβ = B. |a|→1

|a|→1

Therefore, from the definition of the essential norm, we obtain

Dnϕ ,u e,Bα →Bβ = inf Dnϕ ,u − K Bα →Bβ  max{A, B}. K

Next, let {z j } j∈N be a sequence in D such that |ϕ (zj )| → 1 as j → ∞. Define

h j (z) =

1 − |ϕ(z j )|2

(1 − ϕ(z j )z)α

−

α (1 − |ϕ(z j )|2 )2 , α + n (1 − ϕ(z j )z)α+1

−

(1 − |ϕ(z j )|2 )2 . α + n + 1 (1 − ϕ(z j )z)α+1

and

k j (z) =

1 − |ϕ(z j )|2

(1 − ϕ(z j )z)α

α

Similarly to the above we see that both hj and kj belong to B0α and converges to 0 weakly in B α . Moreover,

h(jn) (ϕ(z j )) = 0, |h(jn+1) (ϕ(z j ))| = α(α + 1) · · · (n + α − 1)

|ϕ(z j )|n+1 , (1 − |ϕ(z j )|2 )n+α

and

|k(jn) (ϕ(z j ))| =

|ϕ(z j )|n α(α + 1) · · · (n + α − 1) , k(n+1) (ϕ(z j )) = 0. n+α+1 (1 − |ϕ(z j )|2 )n+α−1 j

Then for any compact operator K : B α → B β , we obtain

Dnϕ ,u − K Bα →Bβ  lim sup Dnϕ ,u (h j )Bβ − lim sup K (h j )Bβ j→∞

j→∞

β (1 − |z j |2 ) |u(z j )||ϕ  (z j )||ϕ(z j )|n+1  lim sup 2 n+α j→∞

and

(1 − |ϕ(z j )| )

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X. Zhu / Applied Mathematics and Computation 274 (2016) 133–142

Dnϕ ,u − K Bα →Bβ  lim sup Dnϕ ,u (k j )Bβ − lim sup K (k j )Bβ j→∞

j→∞

2 β

 lim sup j→∞

|u (z

n (1 − |z j | ) j )|ϕ(z j )| . (1 − |ϕ(z j )|2 )n+α−1

From the definition of the essential norm, we obtain

Dnϕ ,u e,Bα →Bβ = inf Dnϕ ,u − K Bα →Bβ  lim sup K

j→∞

= lim sup |ϕ(z)|→1

β

(1 − |z j |2 ) |u(z j )||ϕ  (z j )| (1 − |ϕ(z j )|2 )n+α

2 β

(1 − |z| ) |u(z)||ϕ  (z)| = E, (1 − |ϕ(z)|2 )n+α

Dnϕ ,u e,Bα →Bβ = inf Dnϕ ,u − K Bα →Bβ  lim sup K

j→∞

β

(1 − |z j |2 ) |u (z j )|ϕ(z j )|n (1 − |ϕ(z j )|2 )n+α−1

β (1 − |z|2 ) |u (z)| = lim sup = F. n+α −1 |ϕ(z)|→1 (1 − |ϕ(z)|2 )

Hence

Dnϕ ,u e,Bα →Bβ  max{E, F }. Now, we prove that

Dnϕ ,u e,Bα →Bβ  max{A, B} and Dnϕ ,u e,Bα →Bβ  max{E, F }. For r ∈ [0, 1), set Kr : H (D) → H (D) by

(Kr f )(z) = fr (z) = f (rz), f ∈ H (D). It is obvious that fr → f uniformly on compact subsets of D as r → 1. Moreover, the operator Kr is compact on B α and Kr Bα →Bα ≤ 1 (see [14]). Let {rj } ⊂ (0, 1) be a sequence such that rj → 1 as j → ∞. Then for all positive integer j, the operator Dnϕ ,u Kr j : B α → B β is compact. By the definition of the essential norm, we get

Dnϕ ,u e,Bα →Bβ ≤ lim sup Dnϕ ,u − Dnϕ ,u Kr j Bα →Bβ .

(2.1)

j→∞

Therefore, we only need to prove that

lim sup Dnϕ ,u − Dnϕ ,u Kr j Bα →Bβ  max{A, B} j→∞

and

lim sup Dnϕ ,u − Dnϕ ,u Kr j Bα →Bβ  max{E, F }. j→∞

For any f ∈ B α such that  f Bα ≤ 1, we consider

(Dnϕ ,u − Dnϕ ,u Kr j ) f Bβ = |u(0) f (n) (ϕ(0)) − rnj u(0) f (n) (r j ϕ(0))| + u · ( f − fr j )(n) ◦ ϕβ .

(2.2)

It is obvious that

lim |u(0) f (n) (ϕ(0)) − rnj u(0) f (n) (r j ϕ(0))| = 0.

j→∞

Now, we consider

lim sup u · ( f − fr j )(n) ◦ ϕβ j→∞

≤ lim sup sup j→∞

|ϕ(z)|≤rN

+ lim sup sup j→∞

|ϕ(z)|>rN

+ lim sup sup j→∞

β

(1 − |z|2 ) |( f − fr j )(n+1) (ϕ(z))||ϕ  (z)||u(z)|

|ϕ(z)|≤rN

β

(1 − |z|2 ) |( f − fr j )(n+1) (ϕ(z))||ϕ  (z)||u(z)| β

(1 − |z|2 ) |( f − fr j )(n) (ϕ(z))||u (z)|

(2.3)

X. Zhu / Applied Mathematics and Computation 274 (2016) 133–142

β

+ lim sup sup

|ϕ(z)|>rN

j→∞

(1 − |z|2 ) |( f − fr j )(n) (ϕ(z))||u (z)|

= Q1 + Q2 + Q3 + Q4 ,

(2.4)

where N ∈ N is large enough such that r j ≥

Q1 := lim sup sup

|ϕ(z)|≤rN

j→∞

|ϕ(z)|>rN

for all j ≥ N,

β (1 − |z|2 ) |( f − fr j )(n+1) (ϕ(z))||ϕ  (z)||u(z)|,

(1 − |z|2 ) |( f − fr j )(n+1) (ϕ(z))||ϕ  (z)||u(z)|, β

Q3 := lim sup sup

|ϕ(z)|≤rN

j→∞

1 2

β

Q2 := lim sup sup j→∞

137

(1 − |z|2 ) |( f − fr j )(n) (ϕ(z))||u (z)|,

and β

Q4 := lim sup sup

|ϕ(z)|>rN

j→∞

(1 − |z|2 ) |( f − fr j )(n) (ϕ(z))||u (z)|.

Since Dnϕ ,u : B α → B β is bounded, applying the operator Dnϕ ,u to zm with m = n, n + 1, we obtain

(Dnϕ ,u zn ) (z) = u (z)n! and (Dnϕ ,u zn+1 ) (z) = u (z)(n + 1)!ϕ(z) + u(z)(n + 1)!ϕ  (z). Thus, u ∈ B β , and using the boundedness of ϕ , we also get

:= sup (1 − |z|2 )β |ϕ  (z)||u(z)| < ∞. K z∈D

Since

(n+1) rn+1 fr j j

→ f (n+1) uniformly on compact subsets of D as j → ∞, we have

lim sup sup | f (n+1) (w) − rn+1 f (n+1) (r j w)| = 0. Q1 ≤ K j j→∞

(2.5)

|w|≤rN

Similarly, from the fact that u ∈ B β we have

Q3 ≤ uBβ lim sup sup | f (n) (w) − rnj f (n) (r j w)| = 0. j→∞

(2.6)

|w|≤rN

Next we consider Q2 . We have Q2 ≤ lim sup j→∞ (S1 + S2 ), where

S1 :=

β

sup

(1 − |z|2 ) | f (n+1) (ϕ(z))||ϕ  (z)||u(z)|

sup

(1 − |z|2 ) rn+1 | f (n+1) (r j ϕ(z))||ϕ  (z)||u(z)|. j

|ϕ(z)|>rN

and

S2 :=

|ϕ(z)|>rN

β

First we estimate S1 . Using the fact that  f Bα ≤ 1, we have

S1 =

sup

|ϕ(z)|>rN

×

β

(1 − |z|2 ) | f (n+1) (ϕ(z))||ϕ  (z)||u(z)|

α +n

(1 − |ϕ(z)|2 ) α(α + 1) · · · (α + n − 1)|ϕ(z)|n+1

α(α + 1) · · · (α + n − 1)|ϕ(z)|n+1 (1 − |ϕ(z)|2 )α+n 1

β

 f Bα sup (1 − |z|2 ) |ϕ  (z)||u(z)| α(α + 1) · · · (α + n − 1)rNn+1 |ϕ(z)|>rN α(α + 1) · · · (α + n − 1)|ϕ(z)|n+1 × (1 − |ϕ(z)|2 )α+n α(α + 1) · · · (α + n − 1)|ϕ(z)|n+1 β  sup sup (1 − |z|2 ) |ϕ  (z)||u(z)| × (1 − |ϕ(z)|2 )α+n |ϕ(z)|>rN |a|>rN    2  n 1 − |a|2 α (1 − |a|2 )    sup  D − ϕ ,u α  α + n (1 − az)α+1 β (1 − az) |a|>rN      2   n  n 1 − |a|2  α (1 − |a|2 )    .   sup D  sup Dϕ ,u + (1 − az)α Bβ α + n |a|>rN  ϕ ,u (1 − az)α+1 Bβ |a|>rN 

(2.7)

138

X. Zhu / Applied Mathematics and Computation 274 (2016) 133–142

Taking limit as N → ∞ we obtain

    n 1 − |a|2    lim sup S1  lim sup Dϕ ,u (1 − az)α  j→∞ |a|→1

Bβ

  2   n (1 − |a|2 )    + lim sup Dϕ ,u |a|→1 (1 − az)α+1 

Bβ

≤ A + B. Similarly, we have lim sup j→∞ S2  A + B, i.e., we get that

Q2  A + B  max{A, B}.

(2.8)

From (2.7), we see that β

lim sup S1  lim sup |ϕ(z)|→1

j→∞

(1 − |z|2 ) |u(z)||ϕ  (z)| = E. (1 − |ϕ(z)|2 )n+α

Similarly we have lim sup j→∞ S2  E. Therefore,

Q2  E.

(2.9)

Next we consider Q4 . We have Q4 ≤ lim sup j→∞ (S3 + S4 ), where

S3 :=

β

sup

(1 − |z|2 ) | f (n) (ϕ(z))||u (z)|

sup

(1 − |z|2 ) rnj | f (n) (r j ϕ(z))||u (z)|.

|ϕ(z)|>rN

and

S4 :=

|ϕ(z)|>rN

β

After some calculation, we have

2n (α + n + 1) β  f Bα sup (1 − |z|2 ) |u (z)| α(α + 1) · · · (α + n − 1) |ϕ(z)|>rN

S3 

×

1 α(α + 1) · · · (α + n − 1)|ϕ(z)|n α +n−1 α+n+1 (1 − |ϕ(z)|2 )

(2.10)

2n (α + n + 1) 2 β sup sup (1 − |z| ) |u (z)| α(α + 1) · · · (α + n − 1) |ϕ(z)|>rN |a|>rN



×

1 α(α + 1) · · · (α + n − 1)|ϕ(z)|n α +n−1 α+n+1 (1 − |ϕ(z)|2 )

  2  2  n α (1 − |a|2 )   Dϕ ,u 1 − |a| α −  α + n + 1 (1 − az)α+1 β (1 − az) |a|>rN      2   n  n 1 − |a|2  α (1 − |a|2 )     sup  sup  D + D  ϕ ,u (1 − az)α  β α + n + 1 |a|>r  ϕ ,u (1 − az)α+1  β |a|>rN N B B       2 2 2    n  n 1 − | a | 1 − | a | ) (   . Dϕ ,u ≤ sup  Dϕ ,u (1 − az)α  β + |sup  |a|>rN a|>rN (1 − az)α+1 Bβ B  sup

Taking limit as N → ∞ we obtain

 

n lim sup S3  lim sup  Dϕ ,u j→∞

|a|→1



   2    n (1 − |a|2 )     + lim sup D  ϕ ,u (1 − az)α+1  β (1 − az)α Bβ |a|→1 B 1 − |a|2

≤ A + B. Similarly, we have lim sup j→∞ S4  A + B, i.e., we get that

Q4  A + B  max{A, B}.

(2.11)

From (2.10), we see that β

lim sup S3  lim sup j→∞

|ϕ(z)|→1

(1 − |z|2 ) |u (z)| = F, (1 − |ϕ(z)|2 )n+α−1

and similarly we have that lim sup j→∞ S4  F . Therefore,

Q4  F.

(2.12)

X. Zhu / Applied Mathematics and Computation 274 (2016) 133–142

139

Hence, by (2.2)–(2.6), (2.8) and (2.11) we get

lim sup Dnϕ ,u − Dnϕ ,u Kr j Bα →Bβ j→∞

= lim sup sup j→∞

 f Bα ≤1

= lim sup sup j→∞

 f Bα ≤1

(Dnϕ ,u − Dnϕ ,u Kr j ) f Bβ u · ( f − fr j )(m) ◦ ϕβ  max{A, B}.

(2.13)

Similarly, by (2.2)–(2.6), (2.9) and (2.12) we get

lim sup Dnϕ ,u − Dnϕ ,u Kr j Bα →Bβ  max{E, F }.

(2.14)

j→∞

Therefore, by (2.1), (2.13) and (2.14), we obtain

Dnϕ ,u e,Bα →Bβ  max{A, B} and Dnϕ ,u e,Bα →Bβ  max{E, F }. This completes the proof of this theorem.  3. New characterization of Dnϕ,u : Bα → Bβ In this section, we give a new characterization for the boundedness, compactness and essential norm of the operator Dnϕ ,u :

B α → B β . For this purpose, we state some definitions and some lemmas which will be used.

Let v : D → R+ be a continuous, strictly positive and bounded function. The weighted space, denoted by Hv∞ , consisting of all f ∈ H (D) such that

 f v = sup v(z)| f (z)| < ∞. z∈D

Hv∞ is a Banach space under the norm  · v . If v(z) = v(|z|) for all z ∈ D, the weighted v is called radial. The associated weight

v of v is defined by

v = ( sup{| f (z)| : f ∈ Hv∞ ,  f v ≤ 1})−1 , z ∈ D. α

When v = vα (z) = (1 − |z|2 ) (0 < α < ∞), it is easy to check that

vα (z) = vα (z). In this case, we denote Hv∞ by Hv∞α , where, α

Hv∞α = { f ∈ H (D) :  f vα = sup | f (z)|(1 − |z|2 ) < ∞}. z∈D

Lemma 3.1. ([4]) For α > 0, we have limk→∞ kα zk−1 vα = ( 2eα )α . Lemma 3.2. ([17]) Let v and w be radial, non-increasing weights tending to zero at the boundary of D. Then the following statements hold. ∞ is bounded if and only if (a) The weighted composition operator uCϕ : Hv∞ → Hw

sup z∈D

w(z) |u(z)| < ∞.

v(ϕ(z))

Moreover, the following holds

uCϕ Hv∞ →Hw∞ = sup z∈D

w(z) |u(z)|.

v(ϕ(z))

∞ is bounded. Then (b) Suppose uCϕ : Hv∞ → Hw

uCϕ e,Hv∞ →Hw∞ = lim− sup

w(z) |u(z)|. v(ϕ(z))

s→1 |ϕ(z)|>s

Lemma 3.3. ([3]) Let v and w be radial, non-increasing weights tending to zero at the boundary of D. Then the following statements hold. ∞ is bounded if and only if (a) uCϕ : Hv∞ → Hw

sup k≥0

u ϕ k  w < ∞, z k  v

with the norm comparable to the above supermum. ∞ is bounded. Then (b) Suppose uCϕ : Hv∞ → Hw

uCϕ e,Hv∞ →Hw∞ = lim sup k→∞

u ϕ k  w . z k  v

140

X. Zhu / Applied Mathematics and Computation 274 (2016) 133–142

Theorem 3.1. Let n be a positive integer, 0 < α , β < ∞, u ∈ H (D) and ϕ be an analytic self-map of D. Then the operator Dnϕ ,u : B α → B β is bounded if and only if

sup jα +n−1 Iu (ϕ j )Bβ < ∞

and

j≥1

sup jα +n−1 Ju (ϕ j−1 )Bβ < ∞.

(3.1)

j≥1

Proof. By Theorem 1 of [39], Dnϕ ,u : B α → B β is bounded if and only if β

sup z∈D

β

(1 − |z|2 ) |u(z)||ϕ  (z)| (1 − |z|2 ) |u (z)| < ∞ and sup < ∞. n+α n+α −1 2 (1 − |ϕ(z)| ) z∈D (1 − |ϕ(z)|2 )

(3.2)

By Lemma 3.2, the first inequality in (3.2) is equivalent to the weighted composition operator uϕ Cϕ : Hv∞α +n → Hv∞ is bounded. β

By Lemma 3.3, this is equivalent to

sup j≥1

uϕ  ϕ j−1 vβ < ∞. z j−1 vα+n

The second inequality in (3.2) is equivalent to the operator uCϕ : Hv∞α +n−1 → Hv∞ is bounded. By Lemma 3.3, this is equivalent to β

u ϕ j−1 vβ sup j−1 < ∞. vα+n−1 j≥1 z Since Iu g(0) = 0, Ju g(0) = 0,

(Iu (ϕ j )(z)) = ju(z)ϕ  (z)ϕ j−1 (z),

( Ju (ϕ j−1 )(z)) = u (z)ϕ j−1 (z),

by Lemma 3.1, we see that Dnϕ ,u : B α → B β is bounded if and only if

jα +n uϕ  ϕ j−1 vβ sup jα +n−1 Iu (ϕ j )Bβ = sup jα +n uϕ  ϕ j−1 vβ ≈ sup α +n j−1 <∞ j z vα+n j≥1 j≥1 j≥1 and

jα +n−1 u ϕ j−1 vβ sup jα +n−1 Ju (ϕ j−1 )Bβ = sup jα +n−1 u ϕ j−1 vβ ≈ sup α +n−1 j−1 < ∞. z vα+n−1 j≥1 j≥1 j≥1 j The proof is complete.  Theorem 3.2. Let n be a positive integer, 0 < α , β < ∞, u ∈ H (D) and ϕ be an analytic self-map of D such that Dnϕ ,u : B α → B β is bounded. Then

Dnϕ ,u e,Bα →Bβ ≈ max{lim sup jα+n−1 Iu (ϕ j )Bβ , lim sup jα+n−1 Ju (ϕ j−1 )Bβ }. j→∞

(3.3)

j→∞

Proof. From the proof of Theorem 3.1 we see that the boundedness of Dnϕ ,u : B α → B β is equivalent to the boundedness of the operators uϕ Cϕ : Hv∞α +n → Hv∞ and uCϕ : Hv∞α +n−1 → Hv∞ . β

β

The upper estimate. By Lemmas 3.1 and 3.3, we get

uϕ Cϕ e,Hv∞α+n →Hv∞β = lim sup j→∞

uϕ  ϕ j−1 vβ jα +n uϕ  ϕ j−1 vβ = lim sup z j−1 vα+n jα +n z j−1 vα+n j→∞

≈ lim sup jα +n uϕ  ϕ j−1 vβ = lim sup jα +n−1 Iu (ϕ j )Bβ j→∞

j→∞

and

uCϕ e,Hv∞α+n−1 →Hv∞β = lim sup j→∞

u ϕ j−1 vβ jα +n−1 u ϕ j−1 vβ = lim sup z j−1 vα+n−1 jα +n−1 z j−1 vα+n−1 j→∞

≈ lim sup jα +n−1 u ϕ j−1 vβ = lim sup jα +n−1 Ju (ϕ j−1 )Bβ . j→∞

j→∞

Using the estimates we have

Dnϕ ,u e,Bα →Bβ  uϕ Cϕ e,Hv∞α+n →Hv∞β + uCϕ e,Hv∞α+n−1 →Hv∞β  max{lim sup jα +n−1 Iu (ϕ j )Bβ , lim sup jα +n−1 Ju (ϕ j−1 )Bβ }. j→∞

j→∞

The lower estimate. From Theorem 2.1, Lemmas 3.1 and 3.2, we have

Dnϕ ,u e,Bα →Bβ  E = uϕ Cϕ e,Hv∞α+n →Hv∞β = lim sup j→∞

uϕ  ϕ j−1 vβ z j−1 vα+n

X. Zhu / Applied Mathematics and Computation 274 (2016) 133–142

141

≈ lim sup jα +n uϕ  ϕ j−1 vβ = lim sup jα +n−1 Iu (ϕ j )Bβ , j→∞

j→∞

and

Dnϕ ,u e,Bα →Bβ  F = uCϕ e,Hv∞α+n−1 →Hv∞β = lim sup j→∞

u ϕ j−1 vβ z j−1 vα+n−1

≈ lim sup jα +n−1 u ϕ j−1 vβ = lim sup jα +n−1 Ju (ϕ j−1 )Bβ . j→∞

j→∞

Therefore,

Dnϕ ,u e,Bα →Bβ  max{lim sup jα+n−1 Iu (ϕ j )Bβ , lim sup jα+n−1 Ju (ϕ j−1 )Bβ }. j→∞

j→∞

This completes the proof.  From Theorem 3.2, we immediately get the following result. Theorem 3.3. Let n be a positive integer, 0 < α , β < ∞, u ∈ H (D) and ϕ be an analytic self-map of D such that Dnϕ ,u : B α → B β is

bounded. Then Dnϕ ,u : B α → B β is compact if and only if

lim sup jα +n−1 Iu (ϕ j )Bβ = 0 and j→∞

lim sup jα +n−1 Ju (ϕ j−1 )Bβ = 0. j→∞

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