Weighted Composition Operators from Analytic Function Spaces into a Class of Weighted-Type Banach Spaces
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2020
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Abstract
This dissertation is under the supervision of Dr. Flavia Colonna, professor of Mathe- matical Sciences at George Mason University. It concerns weighted composition operators from analytic function spaces into a class of weighted-type Banach spaces. In Chapter 2, we characterize the bounded and compact weighted composition opera- tors from a large class of Banach space X of analytic functions on the open unit disk into Zygmund-type spaces Zμ, where μ is a positive continuous function. Under more restric- tive conditions, we provide an approximation of the essential norm of such operators. We also show that all bounded weighted composition operators from X to an important sub- space Zμ,0 of Zμ, called the little Zygmund-type space, are compact and characterize such operators. In Chapter 3, we generalize our work in part one to characterize the bounded and compact weighted composition operators from a large class of Banach space X of analytic functions on the open unit disk into the weighted-type Banach spaces Vn for n ≥ 0. Such spaces have Zμ as a special case for n = 2. The cases when n = 0, 1 have been studied by Colonna and Tjani in 2016. Under more restrictive conditions, we provide an approximation of the essential norm of such operators. We also show that all bounded weighted composition operators from X to the little weighted-type space Vn,0 spaces are compact and characterize such operators. In Chapter 4, we apply our results to the cases when X is the Hardy space Hp for 1≤p≤∞andtheweightedBergmanspaceApα forα>−1and1≤p<∞. Since our general results from Chapters 2 and 3 are not applicable to the case when X is the space Sp of analytic functions whose derivatives are in the Hardy space Hp (for p ≥ 1), independently, we carry out in Chapter 5 the study of the weighted composition operators mapping Sp into the weighted-type Banach spaces Zμ and Vn. This work is motivated by recent studies that have been done by Colonna and Tjani. Indeed, in [9], Colonna and Tjani studied the weighted composition operators from a general reproducing kernel Hilbert space of analytic functions H to the Banach spaces Hμ∞ and Bμ. They characterized the bounded and the compact weighted composition operators from H into Hμ∞ and Bμ. Moreover, they obtained an approximation of the essential norm of such operators. In [10], they extended their results to the weighted composition operators from a large class of Banach space X to the same target spaces Hμ∞ and Bμ. In [8], Colonna and Tjani analyzed the weighted composition operators acting on a general reproducing kernel Hilbert space of analytic functions H but taking as target space the weighted Zygmund space Zμ. They gave characterizations of the boundedness and compactness of such operators and provided an approximation of the essential norm. In [11], they studied the weighted composition operators mapping into Bloch-type spaces when the domain is a large class of Banach spaces X for which the results in the earlier works were not applicable. They also focused on the weighted composition operators acting on the space Sp of the analytic functions on the unit disk whose derivative is in the Hardy space Hp for p ≥ 1. Finally, the boundedness and the compactness of the weighted composition operators into a subspace of the main target space has also been discussed in [9], [10], and [8].