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Journal of Function Spaces
Volume 2017 (2017), Article ID 1409642, 14 pages
https://doi.org/10.1155/2017/1409642
Research Article

Essential Norms of Volterra Type Operators between Zygmund Type Spaces

1School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, China
2Department of Mathematics, Fujian Normal University, Fuzhou 350007, China

Correspondence should be addressed to Shanli Ye; moc.nuyila@ilnahs_ey

Received 5 January 2017; Revised 4 March 2017; Accepted 7 March 2017; Published 11 April 2017

Academic Editor: Ruhan Zhao

Copyright © 2017 Shanli Ye and Caishu Lin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate the boundedness of some Volterra type operators between Zygmund type spaces. Then, we give the essential norms of such operators in terms of , their derivatives, and the nth power of .

1. Introduction

Let be the open unit disk in the complex plane and let be its boundary, and denote the set of all analytic functions on .

For every , we denote by the Bloch type space of all functions satisfying endowed with the norm . The little Bloch type space consists of all satisfying , and is obviously the closed subspace of . When , we get the classical Bloch space and little Bloch space . It is well known that, for , is a subspace of , the Banach space of bounded analytic functions on . Some sources for results and references about the Bloch type functions are the papers of Yoneda [1], Stevic [2, 3], and the first author [47].

For we denote by the Zygmund type space of those functions satisfying and the little Zygmund type space consists of all satisfying , and is obviously the closed subspace of . It can easily be proved that is a Banach space under the norm and that is a closed subspace of . When , we get the classical Zygmund space and the little Zygmund space . It is clear that if and only if .

We consider the weighted Banach spaces of analytic functions endowed with norm , where the weight is a continuous, strictly positive, and bounded function. The weight is called radial, if for all For a weight the associated weight is defined by We notice the standard weights , where , and it is well known that . We also consider the logarithmic weight It is straightforward to show that .

For an analytic self-map of and a function , we define the weighted composition operator as for . Weighted composition operators have been extensively studied recently. It is interesting to provide a function theoretic characterization when and induce a bounded or compact composition operator on various function spaces. Some results on the boundedness and compactness of concrete operators between some spaces of analytic functions one of which is of Zygmund type can be found, for example, in [819].

Suppose that is an analytic map. Let and denote the Volterra type operators with the analytic symbol on , respectively:

If , then is an integral operator. While , then is Cesàro operator. Pommerenke introduced the type operator and characterized the boundedness of between spaces in [20]. More recently, boundedness and compactness of type operators between several spaces of analytic functions have been studied by many authors; one may see [21, 22].

In this paper, we consider the following integral type operators, which were introduced by Li and Stevic (see, e.g., [10, 23]); they can be defined by We will characterize the boundedness of those integral type operators between Zygmund type spaces and also estimate their essential norms. The boundedness and compactness of these operators on the logarithmic Bloch space have been characterized in [22].

Recall that essential norm of a bounded linear operator is defined as the distance from to , the space of compact operators from to , namely, It provides a measure of noncompactness of . Clearly, is compact if and only if .

Throughout this paper, constants are denoted by , they are positive and may differ from one occurrence to the other. The notation means that there are positive constants such that .

2. Boundedness

In order to prove the main results of this paper. We need some auxiliary results.

Lemma 1 (see [8, 13]). For and let be a bounded sequence in which converges to uniformly on compact subsets of . Then .

Lemma 2 (see [8, 13]). For every , where , one has(i) and for every ;(ii) and for ;(iii), for every ;(iv), for every ;(v), for every ;(vi), for every .

Lemma 3 (see [8]). Let and a radial, nonincreasing weight tending to at boundary of , and let the weighted composition operator be bounded.(i) If , then is a compact operator.(ii) (iii) If , then

The following lemma is due to [24, 25].

Lemma 4. Let and be radial, nonincreasing weights tending to zero at the boundary of . Then(i) The weighted composition operator maps into if and only if with norm comparable to the above supermum.(ii)

Lemma 5 (see [26]). For every , one has

Theorem 6. Let be an analytic self-map of and .(i) If , then is a bounded operator if and only if and (ii) If , then is a bounded operator if and only if (iii) If , then is a bounded operator if and only if

Proof. Suppose that is bounded from to . Using the test functions and , we have Since is a self-map, we get that , .
For every and given nonzero , we take the test functions for every . One can show that , and are in , , and . Since , it follows that for all with , we have Then Now we use (14) and Lemma 4 to conclude that which shows that (16) is necessary for all case.
Conversely, suppose that and (16) holds. Assume that . From Lemma 2, it follows thatwhich implies that is bounded. This completes the proof of (i).
Next we will prove (ii). The necessity in condition (17) has been proved above. Fixing with , we take the functionfor , where Then we have by [11]. Let . It follows that Since (17) holds and is bounded, we obtain that Noting and together with (15) and Lemma 4, we conclude that (18) holds.
The converse implication can be shown as in the proof of (i).
Finally we will prove (iii). We have proved that (19) holds above. To prove (20), we take function defined in (23) for every with and obtain that Since is bounded and (19) holds, we obtain that therefore, we deduce that (20) holds by (14) and Lemma 4.
The converse implication can be shown as in the proof of (i).

Theorem 7. Let be an analytic self-map of and .(i) If , then is a bounded operator if and only if and (ii) If , then is a bounded operator if and only if (iii) If , then is a bounded operator if and only if (36) holds and

The proof is similar to that of Theorem 6, and the details are omitted.

Theorem 8. Let be an analytic self-map of and .(i) If , then is a bounded operator if and only if and .(ii) If , then is a bounded operator if and only if and (iii) If , then is a bounded operator if and only if and (iv) If , then is a bounded operator if and only if (v) If , then is a bounded operator if and only if (40) holds and

Proof. Suppose that is bounded from to space.
(i) Case . Using functions and , we obtain Then we obtain that and are necessary for all case.
For the converse implication, suppose that and . For , it follows from Lemma 2 that Then is bounded. This completes the proof of (i).
(ii) Case . We consider the test function defined in (30) for every with . It follows that Since and , we get Then we have On the other hand, from (15) and Lemma 4, we have Hence (39) holds.
The converse implication can be shown as in the proof of (i).
(iii) Case . has been proved above. We take the test function in (23) for every with ; by the same way as (ii), we can obtain that (40) holds.
The converse implication can be shown as in the proof of (i).
(iv) Case . We have proved that (41) holds above. To prove (42), we consider another test function . Clearly and . For every with , it follows that Applying (41) we get Noting and using Lemma 4 and (15), we conclude that (42) holds.
(v) Case . We have proved that (40) holds above. Applying test function in (23) for every with , we have With the same calculation for test function with , then , and we have that Therefore, Since , we conclude that (43) holds.

Theorem 9. Let be an analytic self-map of and .(i) If , then is a bounded operator if and only if and .(ii) If , then