Open Access

New characterizations for the products of differentiation and composition operators between Bloch-type spaces

Journal of Inequalities and Applications20142014:502

https://doi.org/10.1186/1029-242X-2014-502

Received: 29 September 2014

Accepted: 28 November 2014

Published: 12 December 2014

Abstract

We use a brief way to give various equivalent characterizations for the boundedness and the essential norm of the operator C φ D m acting on Bloch-type spaces. At the same time, we use this method to easily get a known characterization for the operator D C φ on Bloch-type spaces.

MSC:47B38, 26A24, 32H02, 47B33.

Keywords

essential normdifferentiationcomposition operatorBloch-type space

1 Introduction and preliminaries

Recently there has been a considerable interest on various product-type operators (see, e.g. [119]), and among them on products of composition and differentiation operators (see, e.g. [1, 2, 4, 5, 712, 1519]). One of the problems of interest is to characterize the boundedness and compactness of the composition operator C φ acting on Bloch-type spaces in terms of the n th power of the analytic self-mapping φ of the unit disk D . Very recently, the first author and Zhou have given the characterizations for the boundedness and the essential norm of the products of differentiation and composition operator C φ D m and D C φ acting on Bloch-type spaces in [9, 10], respectively. Inspired by [20], we present here an easier way to research the corresponding problem. Moreover, by this brief method, we first give new equivalent characterizations for the boundedness and the essential norm of the operator C φ D m , and then we obtain the same results for the operator D C φ as in the paper [10].

Let C denote the unit disk in the complex plane . Denote H ( D ) the space of all holomorphic functions on D and S ( D ) the collection of all holomorphic self-mappings on D . The composition operator C φ is defined by C φ f = f φ for f H ( D ) and φ S ( D ) .

The Bloch space of ν-type
B ν = { f H ( D ) : f B ν = sup z D ν ( z ) | f ( z ) | < }

is a Banach space endowed with the norm | f ( 0 ) | + f B ν , where the weight ν : D R + is a continuous, strictly positive and bounded function.

For the standard weights ν α ( z ) = ( 1 | z | 2 ) α for α > 0 , we denote B ν = B α and
f α = sup z D ( 1 | z | 2 ) α | f ( z ) | .

Similarly, B α is a Banach space under the norm f B α = | f ( 0 ) | + f α . When α = 1 , we get the classical Bloch space . We refer the readers to the book [21] for more information as regards the above spaces.

The weighted Banach space of analytic functions
H ν = { f H ( D ) : f ν = sup z D ν ( z ) | f ( z ) | < }
is a Banach space endowed with the norm ν . The weight ν is called radial, if ν ( z ) = ν ( | z | ) for all z D . For a weight ν, the associated weight ν ˜ ( z ) is defined by
ν ˜ ( z ) = ( sup { | f ( z ) | : f H ν , f ν 1 } ) 1 , z D .
For the standard weights ν α ( z ) = ( 1 | z | 2 ) α ( 0 < α < ), we have ν ˜ α ( z ) = ν α ( z ) . We refer the interested readers to [[22], p.39]. In this case, we denote H ν = H ν α and
f ν α = sup z D ( 1 | z | 2 ) α | f ( z ) | .

Then H ν α is a Banach space under the norm f ν α .

For φ S ( D ) , u H ( D ) , the weighted composition operator u C φ is defined by
u C φ ( f ) = u ( f φ ) , f H ( D ) .

As for u 1 , the weighted composition operator is the usual composition operator C φ . When φ is the identity mapping I, the operator u C I is the multiplication operator M u .

The differentiation operator D is defined by
D f = f , f H ( D ) .
The products of differentiation and composition operators D C φ and C φ D m are defined, respectively, as follows:
D C φ f ( z ) = f ( φ ( z ) ) φ ( z ) , C φ D m f = f ( m ) φ , f H ( D ) , m N .
The essential norm of a continuous linear operator T between two normed linear spaces X and Y is its distance from the compact operators. That is,
T e , X Y = inf { T K : K  is compact } ,

where denotes the operator norm. Notice that T e , X Y = 0 if and only if T is compact, so the estimate on T e , X Y will lead to the condition for the operator T to be compact.

Throughout this paper, C will denote a positive constant, the exact value of which will vary from one appearance to the next. The notations A B , A B , A B mean that there maybe different positive constants C such that B / C A C B , A C B , A C B .

For convenience of the reader we list the results related with our conclusions in this paper.

Theorem A [[9], Theorem 1]

Let 0 < α , β < , m be a nonnegative integer and φ be a holomorphic self-map of the unit disk D . Then C φ D m : B α B β is bounded if and only if
sup n N n α 1 C φ D m I n ( z ) β < .

Theorem B [[9], Theorem 2]

Let 0 < α , β < , m be a nonnegative integer and φ be a holomorphic self-map of the unit disk D . Suppose that C φ D m : B α B β is bounded. Then the estimate for the essential norm of C φ D m : B α B β is
C φ D m e lim sup n n α 1 C φ D m I n ( z ) β ,

where I n ( z ) = z n , z D , n N .

Theorem C [[10], Theorem 2.3]

Let 0 < α , β < , and φ S ( D ) . Then D C φ : B α B β is bounded if and only if
sup n 1 n α I φ ( φ n ) β < and sup n 1 n α J φ ( φ n 1 ) β < .

Theorem D [[10], Theorem 3.5]

Let 0 < α , β < and φ S ( D ) . Suppose that D C φ : B α B β is bounded. Then the estimate for the essential norm of D C φ : B α B β is
max { A 3 2 α + 1 , B 2 α + 1 ( 3 α + 4 ) } D C φ e ( A + B ) ,

where A : = ( e 2 ( α + 1 ) ) α + 1 lim sup n n α I φ ( φ n ) β and B : = ( e 2 α ) α lim sup n n α J φ ( φ n 1 ) β . The definitions of I φ ( φ n ) and J φ ( φ n 1 ) can be found in Section  4.

We would like to point out that the first author and Zhou got the above four theorems by using complex calculations and intricate discussions. In this paper, we will use a brief way to give other equivalent characterizations for the boundedness and the essential norm of C φ D m : B α B β on the unit disk in Section 3. In addition, using this method we will show new proofs of Theorem C and Theorem D in Section 4.

2 Lemmas

In this section we quote some lemmas for our further application. The first lemma is a well-known characterization for B α ( 0 < α < ).

Lemma 2.1 For f H ( D ) , m N and α > 0 . Then
f B α f α j = 0 m 1 | f ( j ) ( 0 ) | + sup z D ( 1 | z | 2 ) α + m 1 | f ( m ) ( z ) | < .

So for f B α , the above lemma implies that f H ν α and more general f ( m + 1 ) H ν α + m . Therefore, theories of the weighted composition operator u C φ : H ν H w play a key role in the proof of our main results. Here we list some lemmas which will be used later.

Lemma 2.2 [[23], Proposition 3.1]

Let ν and w be weights. Then the weighted composition operator u C φ : H ν H w is bounded if and only if
sup z D w ( z ) | u ( z ) | ν ˜ ( φ ( z ) ) < .
Moreover, the following holds:
u C φ H ν H w = sup z D w ( z ) | u ( z ) | ν ˜ ( φ ( z ) ) .

Lemma 2.3 [[23], Theorem 4.4]

Let ν and w be radial, non-increasing weights tending to zero at the boundary of D . Suppose u C φ : H ν H w is bounded. Then
u C φ e , H ν H w lim r 1 sup | φ ( z ) | > r w ( z ) | u ( z ) | ν ˜ ( φ ( z ) ) .

Lemma 2.4 [[24], Theorem 2.4]

Let ν and w be radial, non-increasing weights tending to zero at the boundary of D . Then
  1. (a)
    u C φ : H ν H w is bounded if and only if
    sup n 0 u φ n w z n ν <
     
with the norm comparable to the above supremum.
  1. (b)

    u C φ e , H ν H w = lim sup n u φ n w z n ν .

     

Lemma 2.5 [[22], Lemma 2.1]

For α > 0 , we have lim n ( n + 1 ) α z n ν α = ( 2 α e ) α .

The following criterion for compactness follows from an easy modification of [[25], Proposition 3.11]. Hence we omit the details.

Lemma 2.6 Let 0 < α , β < and T be a linear operator from B α to B β . Then T : B α B β is compact if and only if T : B α B β is bounded and for any bounded sequence { f k } k N in B α which converges to zero uniformly on compact subsets of D , T f k B β 0 as k .

3 Boundedness and essential norm of C φ D m

In this section, we give other equivalent characterizations for the boundedness and the essential norm of the operator C φ D m : B α B β with 0 < α , β < .

Theorem 3.1 Let 0 < α , β < , m N , and φ S ( D ) . Then the following statements are equivalent:
  1. (a)

    C φ D m : B α B β is bounded.

     
  2. (b)
    sup z D ( 1 | z | 2 ) β | φ ( z ) | ( 1 | φ ( z ) | 2 ) α + m < .
    (3.1)
     
  3. (c)
    sup n 1 n α + m φ φ n 1 ν β < .
    (3.2)
     
Proof (a) (b). Suppose that C φ D m : B α B β is bounded. Choose f 1 ( z ) = z m + 1 and
f w ( z ) = 1 | φ ( w ) | 2 ( 1 φ ( w ) ¯ z ) α , w D .
It is easy to verify that f 1 B α and f w B α for w D . By C φ D m f β f α for f B α , we obtain
sup z D ( 1 | z | 2 ) β | φ ( z ) | < ,
and
sup z D ( 1 | z | 2 ) β | φ ( z ) | | φ ( z ) | m + 1 ( 1 | φ ( z ) | 2 ) α + m < .
Then it follows that
sup | φ ( z ) | 1 2 ( 1 | z | 2 ) β | φ ( z ) | ( 1 | φ ( z ) | 2 ) α + m sup z D ( 1 | z | 2 ) β | φ ( z ) | < ,
and
sup | φ ( z ) | > 1 2 ( 1 | z | 2 ) β | φ ( z ) | ( 1 | φ ( z ) | 2 ) α + m sup z D ( 1 | z | 2 ) β | φ ( z ) | | φ ( z ) | m + 1 ( 1 | φ ( z ) | 2 ) α + m < .

That is, (b) holds.

(b) (c). From Lemma 2.2, the condition (b) is a necessary and sufficient condition for the boundedness of weighted composition operator φ C φ : H ν α + m H ν β . Further by Lemma 2.4(a) and Lemma 2.5, the boundedness of the weighted composition operator φ C φ : H ν α + m H ν β is equivalent to the following:
sup n 1 φ φ n 1 ν β z n 1 ν α + m = sup n 1 n α + m φ φ n 1 ν β n α + m z n 1 ν α + m sup n 1 n α + m φ φ n 1 ν β < .
(b) (a). Suppose (b) holds. For every f B α , then it follows from Lemma 2.1 that
C φ D m f β = sup z D ( 1 | z | 2 ) β | f ( m + 1 ) ( φ ( z ) ) φ ( z ) | sup z D ( 1 | z | 2 ) β | φ ( z ) | ( 1 | φ ( z ) | 2 ) α + m f B α < .

Moreover, | C φ D m f ( 0 ) | = | f ( m ) ( φ ( 0 ) ) | f B α ( 1 | φ ( 0 ) | 2 ) α + m 1 . Thus C φ D m f B β < , and hence (a) holds. □

Remark 3.2
  1. (1)

    The relation (a) (b) was essentially proved in a very general result in [18]. For convenience of the reader, we sketch the proof in [18].

     
  2. (2)
    One can easily see that
    sup n N n α 1 C φ D m I n ( z ) β = sup n m + 1 n α 1 n ( n 1 ) ( n m ) φ φ n m 1 ν β = sup k 1 ( k + m ) α ( k + m 1 ) k φ φ k 1 ν β sup k 1 k α + m φ φ k 1 ν β = sup n 1 n α + m φ φ n 1 ν β .
     

Therefore, the characterizations for the boundedness of the operator C φ D m in Theorem 3.1 are equivalent to that in Theorem A.

As an application of Theorem 3.1, we present an example of the bounded operator C φ D m , according to either (3.1) or (3.2).

Example 3.3 Let φ ( z ) = z 2 for z D and β = α + m . Then we study the boundedness of C φ D m : B α B α + m . Firstly, by (3.1), it is clear that
sup z D ( 1 | z | 2 ) α + m | φ ( z ) | ( 1 | φ ( z ) | 2 ) α + m = sup z D ( 1 | z | 2 ) α + m | 2 z | ( 1 | z | 4 ) α + m < .
Secondly, by (3.2) we obtain
sup n 1 n α + m φ φ n 1 ν β = sup n 1 n α + m 2 z z 2 ( n 1 ) ν β = sup n 1 n α + m sup z D ( 1 | z | 2 ) α + m | 2 z z 2 ( n 1 ) | sup n 1 n α + m sup x [ 0 , 1 [ ( 1 x ) α + m x n 1 2 = sup n 1 n α + m ( 1 n 1 2 β + n 1 2 ) α + m ( n 1 2 β + n 1 2 ) n 1 2 = sup n 1 ( β n β + n 1 2 ) α + m ( n 1 2 β + n 1 2 ) n 1 2 < .

From each of these conditions, one sees that C φ D m : B α B α + m is bounded.

Next we estimate the essential norm of the operator C φ D m : B α B β for all 0 < α , β < . Denote B ˜ α = { f B α : f ( 0 ) = 0 } . Let D m + 1 : B α H ν α + m be defined by D m + 1 f = f ( m + 1 ) ( z ) . Then we have D m + 1 f ν m + α f B α for f B ˜ α . Since f ( m + 1 ) H ν α + m when f B α , and further by the equality ( C φ D m f ) = φ f ( m + 1 ) ( φ ) for all f B α , it follows that
C φ D m e , B ˜ α B β φ C φ e , H ν α + m H ν β .
(3.3)

Thus we only need to estimate φ C φ e , H ν α + m H ν β for the upper bound of the essential norm of C φ D m . It is obvious that every compact operator T K ( B ˜ α , B β ) can be extended to a compact operator K K ( B α , B β ) . In fact, for every f B α , f f ( 0 ) B ˜ α , and we can define K ( f ) : = T ( f f ( 0 ) ) + f ( 0 ) , which is a compact operator from B α to B β , due to K ( f k ) has convergent subsequence when { f k } is a bounded sequence. In the following lemma we will use the compact operator K r defined on the space B α by K r f ( z ) = f ( r z ) .

Lemma 3.4 If 0 < α , β < and C φ D m is a bounded operator from B α to B β , then
C φ D m e , B ˜ α B β = C φ D m e , B α B β .
Proof Although the proof is similar to [[20], Lemma 3.1], we will give all the details for convenience of the reader. It is obvious that
C φ D m e , B ˜ α B β C φ D m e , B α B β .
Conversely, let T K ( B α , B β ) be given. Choose an increasing sequence ( r n ) n in ( 0 , 1 ) converging to 1. We denote by A the closed subspace of B α consisting of all constant functions. Then we have
C φ D m T B α B β = sup f B α 1 C φ D m ( f ) T ( f ) B β sup f B α 1 C φ D m ( f f ( 0 ) ) T | B ˜ α ( f f ( 0 ) ) B β + sup f B α 1 C φ D m ( f ( 0 ) ) T ( f ( 0 ) ) B β sup g B ˜ α C φ D m ( g ) T | B ˜ α ( g ) B β + sup h A C φ D m ( h ) T | A ( h ) B β .
Hence
inf T K ( B α , B β ) C φ D m T B α B β inf T K ( B α , B β ) C φ D m T | B ˜ α B ˜ α B β + inf T K ( B α , B β ) C φ D m T | A A B β C φ D m e , B ˜ α B β + lim n C φ D m ( I K r n ) A B β .
Since C φ D m : B α B β is bounded, it follows that
lim n C φ D m ( I K r n ) A B β C lim n I K r n A B β = 0 .

Thus we obtain C φ D m e , B ˜ α B β C φ D m e , B α B β . The proof is finished. □

Thus by Lemma 3.4 and (3.3) it follows that
C φ D m e , B α B β φ C φ e , H ν α + m H ν β .
(3.4)
Theorem 3.5 Let 0 < α , β < , m N , and φ S ( D ) . Suppose that C φ D m : B α B β is bounded. Then
C φ D m e , B α B β lim sup n n α + m φ φ n 1 ν β lim sup | φ ( z ) | 1 ( 1 | z | 2 ) β | φ ( z ) | ( 1 | φ ( z ) | 2 ) α + m .
(3.5)

Proof If φ < 1 , then by [[26], Lemma 3.1], the operator u C φ : B α H μ is compact. The boundedness (compactness) of C φ D m : B α B β is equivalent to the boundedness (compactness) of φ C φ : B α + m H β . In this case, all items in (3.5) are zero.

If φ = 1 , since C φ D m : B α B β is bounded, then the boundedness of φ C φ : H ν α + m H ν β follows from the proof in Theorem 3.1. Thus by (3.4), Lemma 2.4(b), and Lemma 2.5,
C φ D m e , B α B β φ C φ e , H ν α + m H ν β = lim sup n φ φ n 1 ν β z n 1 ν α + m = lim sup n n α + m φ φ n 1 ν β z n 1 ν α + m n α + m lim sup n n α + m φ φ n 1 ν β .
Since φ = 1 , we may choose a sequence { z k } k N D such that | φ ( z k ) | 1 as k . Define
f k ( z ) = 1 | φ ( z k ) | 2 ( 1 φ ( z k ) ¯ z ) α , k N .
It is easy to show that f k B α and converges to zero uniformly on the compact subsets of D as k . Moreover,
f k ( m + 1 ) ( φ ( z k ) ) = α ( α + 1 ) ( α + m ) ( φ ( z k ) ¯ ) m + 1 ( 1 | φ ( z k ) | 2 ) α + m .
Then for every compact operator T : B α B β , by Lemma 2.6, it follows that lim k T f k β = 0 . Thus
C φ D m T B α B β lim sup k C φ D m ( f k ) β lim sup k T f k β = lim sup k C φ D m ( f k ) β lim sup k ( 1 | z k | 2 ) β | f k ( m + 1 ) ( φ ( z k ) ) φ ( z k ) | lim sup k ( 1 | z k | 2 ) β | φ ( z k ) | | φ ( z k ) | m + 1 ( 1 | φ ( z k ) | 2 ) α + m = lim sup k ( 1 | z k | 2 ) β | φ ( z k ) | ( 1 | φ ( z k ) | 2 ) α + m .
Consequently,
C φ D m e , B α B β lim sup | φ ( z ) | 1 ( 1 | z | 2 ) β | φ ( z ) | ( 1 | φ ( z ) | 2 ) α + m .
Since the operator φ C φ : H ν α + m H ν β is bounded, then applying Lemma 2.3, Lemma 2.4(b), and Lemma 2.5, we get
lim sup | φ ( z ) | 1 ( 1 | z | 2 ) β | φ ( z ) | ( 1 | φ ( z ) | 2 ) α + m φ C φ e , H ν α + m H ν β = lim sup n φ φ n 1 ν β z n 1 ν α + m lim sup n n α + m φ φ n 1 ν β .
Thus
lim sup n n α + m φ φ n 1 ν β C φ D m e , B α B β lim sup | φ ( z ) | 1 ( 1 | z | 2 ) β | φ ( z ) | ( 1 | φ ( z ) | 2 ) α + m lim sup n n α + m φ φ n 1 ν β .
Hence
C φ D m e , B α B β lim sup n n α + m φ φ n 1 ν β lim sup | φ ( z ) | 1 ( 1 | z | 2 ) β | φ ( z ) | ( 1 | φ ( z ) | 2 ) α + m .

This completes the proof. □

Remark 3.6
  1. (1)

    The relation C φ D m e , B α B β lim sup | φ ( z ) | 1 ( 1 | z | 2 ) β | φ ( z ) | ( 1 | φ ( z ) | 2 ) α + m can be proved similarly to [[26], Theorem 3.2]. Here we give a complete proof for the reader’s convenience.

     
  2. (2)
    Similar to Remark 3.2, one can get
    lim sup n n α + m φ φ n 1 ν β lim sup n n α 1 C φ D m I n ( z ) β .
     

Therefore, the characterizations for the essential norms of the operator C φ D m in Theorem 3.5 are equivalent to that in Theorem B.

The following corollary is an immediate consequence of Theorem 3.5.

Corollary 3.7 Let 0 < α , β < , m N , and φ S ( D ) . Then the following statements are equivalent:
  1. (a)

    C φ D m : B α B β is compact.

     
  2. (b)
    C φ D m : B α B β is bounded and
    lim sup | φ ( z ) | 1 ( 1 | z | 2 ) β | φ ( z ) | ( 1 | φ ( z ) | 2 ) α + m = 0 .
     
  3. (c)
    C φ D m : B α B β is bounded and
    lim sup n n α + m φ φ n 1 ν β = 0 .
     

4 Boundedness and essential norm of D C φ

In this section, the corresponding problems for the operator D C φ : B α B β are considered. Let u H ( D ) , then for every f H ( D ) , define
I u f ( z ) = 0 z f ( ζ ) u ( ζ ) d ζ , J u f ( z ) = 0 z f ( ζ ) u ( ζ ) d ζ .
Then it follows that
I φ ( φ n ) ( z ) = 0 z ( φ n ) ( ζ ) φ ( ζ ) d ζ , J φ ( φ n 1 ) ( z ) = 0 z φ n 1 ( ζ ) φ ( ζ ) d ζ .
By an easy calculation, one can get
( I φ ( φ n ) ( z ) ) = n φ ( z ) n 1 ( φ ( z ) ) 2
(4.1)
and
( J φ ( φ n 1 ) ( z ) ) = φ ( z ) n 1 φ ( z ) .
(4.2)

In 2007, S Li and S Stević gave the following characterizations for the boundedness and compactness of the operator D C φ : B α B β .

Lemma 4.1 Let α , β > 0 and φ S ( D ) . Then the following statements hold:
  1. (a)
    [[4], Theorem  1] D C φ : B α B β is bounded if and only if
    sup z D | φ ( z ) | 2 ( 1 | z | 2 ) β ( 1 | φ ( z ) | 2 ) α + 1 < and sup z D | φ ( z ) | ( 1 | z | 2 ) β ( 1 | φ ( z ) | 2 ) α < .
    (4.3)
     
  2. (b)
    [[4], Theorem  2] D C φ : B α B β is compact if and only if D C φ : B α B β is bounded,
    lim | φ ( z ) | 1 | φ ( z ) | 2 ( 1 | z | 2 ) β ( 1 | φ ( z ) | 2 ) α + 1 = 0 and lim | φ ( z ) | 1 | φ ( z ) | ( 1 | z | 2 ) β ( 1 | φ ( z ) | 2 ) α = 0 .
     

First, we will give a brief proof of Theorem C as regards the bounded operator D C φ : B α B β for all 0 < α , β < .

Theorem 4.2 Let 0 < α , β < and φ S ( D ) . Then D C φ : B α B β is bounded if and only if
sup n 1 n α I φ ( φ n ) β < and sup n 1 n α J φ ( φ n 1 ) β < .
Proof Lemma 4.1 shows that D C φ maps B α boundedly into B β if and only if (4.3) holds. On the other hand, Lemma 2.2 shows that (4.3) holds if and only if the weighted composition operators ( φ ) 2 C φ maps H ν α + 1 boundedly into H ν β and φ C φ maps H ν α boundedly into H ν β , and hence it follows from Lemma 2.4(a) that (4.3) is equivalent to
sup n 1 ( φ ) 2 φ n 1 ν β z n 1 ν α + 1 < and sup n 1 φ φ n 1 ν β z n 1 ν α < .
Using Lemma 2.5, (4.1) and (4.2), then the boundedness of D C φ : B α B β is equivalent to
sup n 1 ( φ ) 2 φ n 1 ν β n α + 1 n α + 1 z n 1 ν α + 1 sup n 1 ( φ ) 2 φ n 1 ν β n α + 1 = sup n 1 n α I φ ( φ n ) β <
and
sup n 1 n α φ φ n 1 ν β n α z n 1 ν α sup n 1 n α φ φ n 1 ν β = sup n 1 n α J φ ( φ n 1 ) β < .

This completes the proof. □

Now, we give a new proof of Theorem D about the essential norm of D C φ : B α B β for 0 < α , β < . We denote B ˜ α = { f B α : f ( 0 ) = 0 } . Let D α : B α H ν α and S α : B α H ν α + 1 be the first-order derivative operator and the second-order derivative operator, respectively. That is,
D α ( f ) = f , S α ( f ) = f .
By Lemma 2.1 we have
D α f ν α = f B α and S α f ν α + 1 f B α for  f B ˜ α .
For f B α , by Lemma 2.1, f H ν α + 1 , and f H ν α . Then by the equation ( D C φ f ) = f ( φ ) ( φ ) 2 + f ( φ ) φ , it follows that
D C φ e , B ˜ α B β ( φ ) 2 C φ e , H ν α + 1 H ν β + φ C φ e , H ν α H ν β .
(4.4)
Moreover, every compact operator T K ( B ˜ α , B β ) can be extended to a compact operator K K ( B α , B β ) . Then similar to Lemma 3.4, one can easily get
D C φ e , B ˜ α B β = D C φ e , B α B β .
Thus combining the above equation with (4.4), we obtain
D C φ e , B α B β ( φ ) 2 C φ e , H ν α + 1 H ν β + φ C φ e , H ν α H ν β .
(4.5)

According to (4.5), we only need to estimate the right two essential norms for the upper bound of the essential norm of D C φ : B α B β .

Theorem 4.3 Let 0 < α , β < and φ S ( D ) . Suppose that D C φ : B α B β is bounded. Then
D C φ e , B α B β max { lim sup n n α I φ ( φ n ) β , lim sup n n α J φ ( φ n 1 ) β } .

Proof By Lemma 4.1(a) and Lemma 2.2, the boundedness of D C φ : B α B β is equivalent to ( φ ) 2 C φ : H ν α + 1 H ν β and φ C φ : H ν α H ν β are bounded weighted composition operators.

The upper estimate. From Lemma 2.4(b) and Lemma 2.5, we obtain
( φ ) 2 C φ e , H ν α + 1 H ν β = lim sup n ( φ ) 2 φ n 1 ν β z n 1 ν α + 1 = lim sup n n α + 1 ( φ ) 2 φ n 1 ν β n α + 1 z n 1 ν α + 1 lim sup n n α + 1 ( φ ) 2 φ n 1 ν β = lim sup n n α I φ ( φ n ) β , φ C φ e , H ν α H ν β = lim sup n φ φ n 1 ν β z n 1 ν α = lim sup n φ φ n 1 ν β n α z n 1 ν α n α lim sup n n α φ φ n 1 ν β = lim sup n n α J φ ( φ n 1 ) β .
Then it follows from (4.5) that
D C φ e , B α B β max { lim sup n n α I φ ( φ n ) β , lim sup n n α J φ ( φ n 1 ) β } .
The lower estimate. Let { z k } k N be a sequence in D such that | φ ( z k ) | 1 as k . Define
f k ( z ) = 1 | φ ( z k ) | 2 ( 1 φ ( z k ) ¯ z ) α α α + 1 ( 1 | φ ( z k ) | 2 ) 2 ( 1 φ ( z k ) ¯ z ) α + 1 , g k ( z ) = 1 | φ ( z k ) | 2 ( 1 φ ( z k ) ¯ z ) α α α + 2 ( 1 | φ ( z k ) | 2 ) 2 ( 1 φ ( z k ) ¯ z ) α + 1 .
We can easily show both f k and g k belong to B α and converge to zero uniformly on the compact subsets of D as k . Moreover,
f k ( φ ( z k ) ) = 0 , f k ( φ ( z k ) ) = α ( φ ( z k ) ¯ ) 2 ( 1 | φ ( z k ) | 2 ) α + 1 ; g k ( φ ( z k ) ) = α φ ( z k ) ¯ ( α + 2 ) ( 1 | φ ( z k ) | 2 ) α , g k ( φ ( z k ) ) = 0 .
Then for every compact operator T : B α B β , by Lemma 2.6 we obtain
D C φ T B α B β lim sup k D C φ ( f k ) β lim sup k ( 1 | z k | 2 ) β | α ( φ ( z k ) ) 2 ( φ ( z k ) ¯ ) 2 ( 1 | φ ( z k ) | 2 ) α + 1 | , D C φ T B α B β lim sup k D C φ ( g k ) β lim sup k ( 1 | z k | 2 ) β | α φ ( z k ) φ ( z k ) ¯ ( α + 2 ) ( 1 | φ ( z k ) | 2 ) α | .
Since the weighted composition operators ( φ ) 2 C φ : H ν α + 1 H ν β and φ C φ : H ν α H ν β are bounded. Then applying Lemma 2.3, Lemma 2.4(b), and Lemma 2.5, it follows that
D C φ e , B α B β lim sup | φ ( z ) | 1 ( 1 | z | 2 ) β | ( φ ( z ) ) 2 ( 1 | φ ( z ) | 2 ) α + 1 | ( φ ) 2 C φ e , H ν α + 1 H ν β = lim sup n ( φ ) 2 φ n 1 ν β z n 1 ν α + 1 lim sup n n α + 1 ( φ ) 2 φ n 1 ν β = lim sup n n α I φ ( φ n ) β , D C φ e , B α B β lim sup | φ ( z ) | 1 ( 1 | z | 2 ) β | φ ( z ) ( 1 | φ ( z ) | 2 ) α | φ C φ e , H ν α H ν β = lim sup n φ φ n 1 ν β z n 1 ν α lim sup n n α φ φ n 1 ν β = lim sup n n α J φ ( φ n 1 ) β .
Hence
D C φ e , B α B β max { lim sup n n α I φ ( φ n ) β , lim sup n n α J φ ( φ n 1 ) β } .

This completes the proof. □

The following result is an immediate consequence of Theorem 4.3 and Lemma 4.1(b).

Corollary 4.4 Let α , β > 0 and φ S ( D ) . Then the following statements are equivalent:
  1. (a)

    D C φ : B α B β is compact.

     
  2. (b)
    D C φ : B α B β is bounded,
    lim sup n n α I φ ( φ n ) β = 0 and lim sup n n α J φ ( φ n 1 ) β = 0 .
     

Declarations

Acknowledgements

The authors would like to thank the Journal Editorial Office and the referees for the useful comments and suggestions, which improved the presentation of this article. This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11201331; 11301373; 11401431).

Authors’ Affiliations

(1)
School of Mathematical Sciences, Tianjin Normal University
(2)
Department of Mathematics, Tianjin University

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© Liang and Dong; licensee Springer. 2014

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