Let the holomorphic mapping ?, and the holomorphic self-map ? are on the upper half-plane. We characterize bounded weighted composition operators between the Hardy space and the weighted-type space on the upper half-plane, and we study the special cases when which is the Hilbert space. Under a mild condition on ?; we also show the compactness of these operators and there special cases. Keywords: Weighted composition operators, Hardy spaces, weighted type spaces, upper half plane

Elniel, Elhadi Elnour

English

Let the holomorphic mapping ψ, and the holomorphic self-map φ are on the upper half-plane. We characterize bounded weighted composition operators between the Hardy space and the weighted-type space on the upper half-plane, and we study the special cases when which is the Hilbert space. Under a mild condition on ψ; we also show the compactness of these operators and there special cases

Elhadi Elnour Elniel

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Weighted Composition Operators from Hardy Spaces to Weighted-Type Spaces on the Upper Half-Plane

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Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.4, No.14, 2014  40  Weighted Composition Operators from Hardy Spaces to Weighted-Type Spaces on the Upper Half-Plane Elhadi Elnour Elniel  Department of Mathematics, College of Arts and Science,  Al Mandag , Al Baha University, Saudi Arabia elhadielniel_2003@hotmail.com  Abstract     Let the holomorphic mapping ψ, and the holomorphic self-map φ are on the upper half-plane. We characterize bounded weighted composition operators between the Hardy space and the weighted-type space on the upper half-plane, and we study the special cases when  which is the Hilbert space. Under a mild condition on ψ; we also show the compactness of these operators and there special cases.  Keywords: Weighted composition operators, Hardy spaces, weighted type spaces, upper half plane.  1. Introduction Let      be the upper half-plane,  a domain in  or , and the space of all holomorphic functions on   . Let  , and let  be a holomorphic self-map of . Then by   (1) is defined a linear operator on  which is called weighted composition operator. If  then  becomes composition operator and is denoted by , and if   then   becomes multiplication operators and is denoted by   . During the past few decades, composition operators and weighted composition operators have been studied extensively on spaces of holomorphic functions on various domains in  or   (see, e.g. . For some other operators related to weighted composition operators, see [22-29].  While there is a vast literature on composition and weighted composition operators between spaces of holomorphic functions on the unit disk , there are few papers on these and related operators on spaces of functions holomorphic in the upper half-plane ( . For related results in the setting of the complex plane see also papers . The behavior of composition operators on spaces of functions holomorphic in the upper half-plane is considerably different from the behavior of composition operators on spaces of functions holomorphic in the unit disk . For example, there are holomorphic self-maps of  which do not induce composition operators on Hardy and Bergman spaces on the upper half-plane, where as it is a well-known consequence of the Little wood subordination principle that every holomorphic self-map  of  induces a bounded composition operator on the Hardy and weighted Bergman spaces on . Also, Hardy and Bergman spaces on the upper half-plane do not support compact composition operators (see ).  For   and    let    denote the collection of all Lebesgue p-integrable functions    such that :  (2) where  (3) Let   . For ,  is a Banach space with the norm defined by  Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.4, No.14, 2014  41 with this norm  becomes a Banach space when  while for  it is a Frechet space with the translation invariant metric Recall that for every the following estimate holds: where  is a positive constant independent of   Let . The weighted-type space (or growth space) on the upper half-plane consists of all  such that  For weighted type spaces on the unit disk, polydisk, or the unit ball see, for example, paper . Given two Banach spaces  and  we recall that a linear map   is bounded if    is bounded for every bounded subset  of . In addition, we say that  is compact if    is relatively compact for every bounded set E Y. In this paper we follow the same Literature and methods of Stevo Stevic, Ajay K. Sharma, and S.D. Sharma [29] with a little change.  We consider the boundedness and compactness of weighted composition operators acting from  to the weighted-type space  . Throughout this paper, constants are denoted by ; they are positive and may different from one occurrence to the other.   2. Main Results The boundedness and compactness of the weighted composition operators from the Hardy space to the weighted type space on the upper half plane , and the special cases when  which is a Hilbert space are characterize in this section.  Theorem 2.1. Let   ,  , and let  be a holomorphic self-map of . Then  is bounded if and only if  Moreover, if the operator  is bounded then the following asymptotic relationship holds Proof. First suppose that (8) holds. Then for any  and , by(6) we have  and so by ( 8)  is bounded and moreover  Conversely suppose   is bounded. Consider the function  (4)  (5)  (6)  (7)  (8)  (9)  (10)  (11) Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.4, No.14, 2014  42 Then , and moreover  (see, e.g., Lemma1 in [18]). Thus the boundedness of  implies that for every  . In particular, if    is fixed then for , we get  Since is arbitrary,(8) follows and moreover  If   is bounded then from  and  asymptotic relationship (9) follows.                    Corollary 2.2. Let   be such that  and . Then  is bounded if and only if   , where Example 2.3. Let   and   be such that  and . Let  be a holomorphic map of  defined as  For    and    in  , we have   Thus  if    Similarly  if . By Corollary 2.2 it follows that  is bounded.  Corollary 2.4. Let   and let  be a holomorphic self-map of   . Then  is bounded if and only if   Corollary 2.5. Let    be the linear fractional map Then necessary and sufficient condition that   is bounded is that   and  .  (12)  (13)  (14)  (15)  (16)  (17)  (18)  (19)  (20) Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.4, No.14, 2014  43 Proof. Assume that   is bounded. Then   which is finite only if   and   .  Conversely, if   and   then from  we get , and by some calculation   Hence that  is bounded.                                                                                                        Corollary 2.6.  Let  and     be such that  Let  be a holomorphic self-map of   and    . Then the weighted composition operator   acts boundedly from   to . Proof. By Theorem 2.1  is bounded if and only if  By the Schwarz-Pick theorem on the upper half-plane we have that for every holomorphic self-map   of  and all  where the equality holds when   is  a Mobius transformation given by(20 ). From (24) , condition (23) follows and consequently the boundedness of the operator   Corollary 2.6. Enables us to show that there exist   > 0,  and  holomorphic maps  and  of the upper half-plane such that neither   nor  is bounded, but   bounded.                                                                                                                               Example 2.7. Let   and    be such that   . Let  ad ,  and  . Then by Corollary 2.5 ,  is not bounded. On the other hand, if  then  and so by Corollary 2.2,  is not bounded. However, by Corollary 2.6, we have that  is bounded.         The next Schwartz-type lemma characterizes compact weighted composition operators   and it follows from standard arguments .  Lemma 2.8. Let ,  and  let   be a holomorphic self-map of  . Then  is compact if and only if, for any bounded sequence  converging to zero on compacts of   , one has  Remark 2.9. Let , , and  let   be a holomorphic self-map of  . Then   (i)  is bounded if and only if   , where  (21)  (22)  (23)  (24)  (25)  (26) Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.4, No.14, 2014  44  (ii)  is bounded if and only if  (iii) The weighted composition operator   acts boundedly from   to  where  . (iv)  is compact if and only if, for any bounded sequence  converging to zero on compacts of  , one has  Theorem 2.10  . Let  and    be a holomorphic self-map of   if   is compact, then Proof. Suppose  is compact and (30) does not hold. Then there is  and a sequence such that  and   for all   Let  , and  Then   is a norm bounded sequence and   on compacts of  as . By Lemma 2.8 it follows that  On the other hand, which is a contradiction. Hence (30) must hold, as claimed.                                                                                                 Before we formulate and prove a converse of Theorem 2.10, we define, for every   such that   the following subset of   :  Theorem 2.11. Let ,  , and let  be a holomorphic self-map of  and   be  bounded. Suppose that   and  as   within  for all a and b, Then   is compact if condition holds.    (27)  (28)  (29)  (30)  (31)  (32)  (33)    (34)  (35) Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.4, No.14, 2014  45 Proof. Assume holds. Then for each ,  there  is  an   such that   Let  be a sequence in  such that    and    uniformly on compact subsets of  as  Thus for   such that   and each  , we have From estimate we have Thus there is an  such that whenever  . Hence for  such that   and each    we have  If , then by the assumption there is an  such that whenever Therefore, for each  we have  whenever  If , then there exists some   such that   for all  , and so Combining - , we have that  for   and some    independent of  . Since  is an arbitrary positive number, by Lemma 2.8, it follows that   is compact.                                                                                                                Example 2.12. and   be such that    Let   and    , then    and  . It is easy to see that . Beside this, for  we have Also and the set  is empty. Thus  and   satisfy all the assumptions of Theorem 2.11, and so  is compact.   Acknowledgment: The author would like to express their thanks to Dr. Mohamed Abdel-Halim El-Sayed and Dr. Omer Othman Ibrahim for the useful help. Also, the author would like to thank the reviewers for their comments, which have greatly assisted them in improving the quality of the presented work.   whenever     (36)    (37)    (38)    (39)    (40)    (41)    (42)    (43)  as       (44)    (45) Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.4, No.14, 2014  46   References 1. Mohammad Ali Ardalan, “Boundedness of Composition Operator Between Weighted Spaces of Holomorphic Functions on The Upper Half-Plane” Taiwanese Journal of Mathematics, Vol. 18, No. 1, pp. 277-283, February 2014. 2. Elke Wolf, "Composition followed by differentiation between weighted Bergman spaces and weighted Banach spaces of holomorphic functions," Acta Universitatis Sapientiae, Mathematica. Volume 6, Issue 1, Pages 107–116, November 2014 3. 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Weighted Composition Operators from Hardy Spaces to Weighted-Type Spaces on the Upper Half-Plane

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