New Characterizations for Weighted Composition Operators from Weighted Bergman Space into n th Weighted-Type Spaces
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RESEARCH PAPER
New Characterizations for Weighted Composition Operators from Weighted Bergman Space into nth Weighted-Type Spaces Xiangling Zhu1
•
Ebrahim Abbasi2
Received: 26 March 2020 / Accepted: 29 July 2020 Ó Shiraz University 2020
Abstract In this work, several new characterizations are given for the boundedness and compactness of weighted composition operators from weighted Bergman spaces into nth weighted-type spaces. Keywords Weighted composition operator Weighted Bergman space nth weighted-type space Mathematics Subject Classification 47B33 32A36 30H99
1 Introduction Let D be the open unit disk in the complex plane C and HðDÞ be the set of all analytic functions on D. Let 1 p\1, a [ 1 and dAa ðzÞ ¼ ð1 þ aÞð1 jzj2 Þa dAðzÞ, where dA is the normalized area measure on D. The weighted Bergman space, denoted by Apa , is the space of all analytic functions on D such that Z jf ðzÞjp dAa ðzÞ\1: kf kpAp ¼ a
D
It is a Banach space with the norm k kApa . A positive and continuous function on D is called a weight. Let l be a weight and n 2 N0 , the set of nonnegative integers. The nth weighted-type space W nl ðDÞ ¼ W nl , which introduced by Stevic´ (2009), consists of all f 2 HðDÞ such that
& Ebrahim Abbasi [email protected]; [email protected] Xiangling Zhu [email protected] 1
University of Electronic Science and Technology of China, Zhongshan Institute, Zhongshan 528402, Guangdong, People’s Republic of China
2
Department of Mathematics, Mahabad Branch, Islamic Azad University, Mahabad, Iran
bW nl ðf Þ :¼ sup lðzÞjf ðnÞ ðzÞj\1: z2D
This space is a Banach space with the following norm kf kW nl ¼
n1 X
jf ðiÞ ð0Þj þ bW nl ðf Þ:
i¼0
When n ¼ 0, W nl becomes the weighted-type space Hl1 and for n ¼ 1 and n ¼ 2, W nl becomes the weighted Bloch space Bl and the weighted Zygmund space Z l , respectively. We refer the interested reader to Abbasi et al. (2019, 2020), Stevic´ (2009, 2010a, b), Zhu and Du (2019) for more information about nth weighted-type spaces. Let a [ 0 and lðzÞ ¼ ð1 jzj2 Þa . For n ¼ 1; 2, the space n W l is called, the Bloch type space Ba and the Zygmund type space Z a , respectively. When a ¼ 1, Ba is the Bloch space B. For more information about Bloch type spaces or Zygmund type spaces, see Zhu (1993, 2005). Let u 2 HðDÞ and u be an analytic self-map of D. The weighted composition operator induced by u and u, denoted by uCu , is defined as follows. ðuCu f ÞðzÞ ¼ uðzÞf ðuðzÞÞ;
f 2 HðDÞ;
z 2 D:
When u 1, then uCu is denoted by the symbol Cu and it is called the composition operator. If uðzÞ ¼ z, then uCu is called the multiplication operator and denoted by Mu . It is important to give function theoretic descriptions of when u and u induce a bounded or compact weighted composition operator on various function spaces. See Cowen and MacCluer (1995) for more details about this topic.
123
Iran J Sci Technol Trans Sci
For n; k 2 N0 with k n, the partial Bell polynomials is defined by Bn;k ðx1 ; x2 ; . . .;
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