Abstract

Let be a weight function such that is in the class of Békollé weights, a normal weight function, a holomorphic map on , and a holomorphic self-map on . In this paper, we give upper and lower bounds for essential norm of weighted composition operator acting from weighted Bergman spaces to Bloch-type spaces

1. Introduction and Preliminaries

Let be the open unit disk in the complex plane and the space of all holomorphic functions on . For a and a holomorphic self-map of , the weighted composition operator is a linear operator on defined by , . Several authors have studied these weighted composition operators on different spaces of analytic functions, see for example, [1–12] and the related references therein. Recently, Stevic and Sharma [12] characterized boundness and compactness of acting from weighted Bergman spaces to Bloch-type spaces Motivated by results in [12], in this paper, we give upper and lower bounds for essential norm of a weighted composition operator acting between these spaces.

A continuous function is called a weight or a weight function. We extend it on by defining for all . For and a weight, denoted by the weighted Bergman space consisting of holomorphic functions on such that where is the normalized area measure in If , then is the well-known weighted Bergman space .

For and , the class of Békollé weights consists of weights with the property that there exists a constant such that

Here, is the probability measure on is the Carleson square in , and is the conjugate exponent of , that is, Recall that a weight is normal if there exist positive numbers and , , and such that

It is well known that classical weights are normal weights.

For a normal weight , the weighted Bloch-type space on is the space of all functions in such that . The space is a Banach space with the norm

Throughout this paper, is fixed, , , and . We also assume that , a weight function such that belongs to , , and be the reproducing kernel of the Bergman space Constants are denoted by ; they are positive and not necessarily the same at each occurrence. The notation means that is less than or equal to a constant multiple of , and means that a constant multiple of is greater than or equal to . When as well as , then we write .

2. Essential Norm of

In this section, we give upper and lower bounds for the essential norm of weighted composition operator

Recall that if and are two Banach spaces, then the essential norm of a bounded linear operator is defined as where denotes the usual operator norm. Clearly, is compact if and only if

Theorem 1. Let , and be a holomorphic self-map of such that . Assume that is bounded. Then, where .

To prove the main result of this paper, we need the following lemmas. The next two lemmas can be found in [12].

Lemma 2. The following estimates hold: (1)For each we have that(2)For each we have thatwhere

Lemma 3. For each the function defined as is in Moreover, and converges to zero, uniformly on compact subsets of as

The next lemma can be found in [8].

Lemma 4. Let . If a bounded sequence in converges to uniformly on compact subsets of then also converges to weakly in .

Proof of Theorem 1. Lower bound. Let be a sequence in such that as and For each let be defined as Then, and Consider the family of functions defined as where is defined as in (9). Also, by Lemma 3, and converges to zero uniformly on compact subsets of as By Lemma 4, converges to zero weakly in Thus, for any compact operator , we have that as Moreover, Also, and Now Therefore, from (12) and (13), we have that Using the facts thatand
we have that Again, let be a sequence in such that as and For each let be defined as Then, by Lemma 3, and converges to zero uniformly on compact subsets of as By Lemma 4, converges to zero weakly in Thus, for any compact operator , we have that as Moreover, and Thus, using (18), we have that Combining (15) and (19), we have that Upper bound. Let be such that . Let where
Then, by Theorem 6.1 in [3], we have that is compact. Since is bounded, so is compact. Thus, for fixed in we have that where is the identity operator on . Now Let and Let be the line segment from to . Then, where . Thus, by Lemma 2, we have that Again, let Then, by Lemma 2, we have that Thus, Thus, from (24) and (26), we have that Similarly, we can show that Also, by Lemma 2 and equation (26), we have that Since is bounded, so for each By taking, respectively, and and using the fact that we have that Combining (23) and (27)-(29), we have that Using (28), we have that Using (31), (32), and (33), we have that as The last term in the right-hand side of (22) is dominated by which is further dominated by a constant multiple of Letting in (36), we get Using (34) and (37) in (22), we have that Finally, letting , then we get Combining (20) and (39), we get the desired result.