- Open Access
General Toeplitz operators on weighted Bloch-type spaces in the unit ball of
© El-Sayed Ahmed; licensee Springer. 2013
- Received: 17 October 2012
- Accepted: 17 April 2013
- Published: 9 May 2013
In this paper, we consider the weighted Bloch-type spaces with and in the unit ball of . We present some basic properties of the spaces , then we consider the Toeplitz operator acting between spaces, where μ is a positive Borel measure in the unit ball . Moreover, we characterize complex measures μ for which the Toeplitz operator is bounded or compact on .
MSC: 47B35, 32A18.
- Toeplitz operators
- weighted Bloch-type spaces
- weighted Bergman spaces
We start here with some terminology, notations and definitions of various classes of analytic functions defined on the unit ball of .
Thus , where 1 stands for a constant function.
Toeplitz operators have been studied extensively on the Bergman spaces by many authors. For references, see  and . Boundedness and compactness of the general Toeplitz operators on the α-Bloch spaces have been investigated in  on the unit disk for . Also, in , the authors extended the general Toeplitz operator to with . Recently, in , the general Toeplitz operators on the analytic Besov spaces with have been investigated. Under a prerequisite condition, the authors characterized a complex measure μ on the unit disk for which is bounded or compact on the Besov space . For more studies on the Toeplitz operator, we refer to [11–17].
In this paper, we consider the weighted Bloch-type spaces with and in the unit ball of . We prove a certain integral representation theorem that is used to determine the degree of growth of the functions in the space . It is also proved that the space is a Banach space for each weight , , and the Banach dual of the Bergman space is for each , . Further, we extend the Toeplitz operator to in the unit ball of and completely characterize the positive Borel measure μ such that is bounded or compact in spaces with .
Throughout the paper, we say that the expressions A and B are equivalent, and write whenever there exist positive constants and such that . As usual, the letter C denotes a positive constant, possibly different on each occurrence. Hereafter, ω stands for a right-continuous and nondecreasing function.
Theorem 1.1 (see [, Theorem 1.12])
have the following asymptotic properties.
(1) If , then and are both bounded in .
Lemma 1.1 (see [, Lemma 3.3])
Let be the Bergman metric on . Denote the Bergman metric ball at by , where and .
Lemma 1.2 (see [, Theorem 2.23])
For fixed , there is a sequence such that:
• there is a positive integer N such that each is contained in at most N of the sets .
then μ is called a vanishing Carleson measure.
If , , then we get the α-Bloch space and the little α-Bloch space . If , and , then we get the classical Bloch space and . These classes extend the weighted Bloch spaces defined in  to the setting of several complex variables.
If and , then we get the logarithmic α-Bloch space and the little logarithmic α-Bloch space . If , and , then we get the logarithmic Bloch space and (see ).
In this section, we study the general -Bloch space in the unit ball of by giving some characterizations of -Bloch space, then we present several auxiliary results, which play important roles in the proofs of our main results.
from which the result follows. □
Theorem 2.1 For each , and . Then the following conditions are equivalent:
(ii) The function is bounded in ;
This proves that (i) ⇒ (ii).
As in the proof of Theorem 7.1 in , we have .
This shows that (ii) implies (iii). That (iii) implies (i) follows from differentiating under the integral sign and then applying Theorem 1.12 in . □
It remains for us to show that .
This shows that and completes the proof of the theorem. □
is bounded in .
It is easy to check that .
is bounded in . This completes the proof. □
Lemma 2.3 Let . Let λ be any real number satisfying the following properties:
• if ;
• if ;
• if .
This proves the necessity. The proof of the sufficiency condition is much akin to the corresponding one in , so the proof is omitted. □
Proposition 2.1 Suppose and . Let λ be any real number satisfying:
• if ;
• if .
In this section, we study the boundedness of general Toeplitz operators acting on the weighted Bloch-type spaces in the unit ball of .
Theorem 3.1 Let μ be a positive Borel measure on . Then we have
(1) if , then is bounded on if and only if and μ is a Carleson measure;
(2) if , then is bounded on if and only if and μ is a Carleson measure;
(3) if , , then is bounded on if and only if and μ is a Carleson measure.
for all and , where C is a positive constant that does not depend on f or g.
It is easy to see that
This implies that . Hence, is a bounded operator on with , .
Hence μ is a Carleson measure on .
By (8), then and .
When , taking , we have .
By (8) it is obvious that .
This completes the proof of Theorem 3.1. □
In this section, we study the compactness of Toeplitz operators on the weighted Bloch-type spaces in the unit ball of . We need the following lemma.
whenever is a bounded sequence in that converges to 0 uniformly on .
Proof This lemma can be proved by Montel’s theorem and Lemma 2.1. □
Theorem 4.1 Let μ be a positive Borel measure on . We have the following:
(1) if , then is compact on if and only if and μ is a vanishing Carleson measure;
(2) if , then is compact on if and only if and μ is a vanishing Carleson measure;
(3) if , , then is compact on if and only if and μ is a vanishing Carleson measure.
From (8) it is easy to see that
which implies that is a compact operator.
This implies that μ is a vanishing Carleson measure on .
with , we have