Some Linear Operators On Certainfunction Spaces

In this thesis, first, we discuss the boundedness and compactness of the product of extended Ces(?)ro and composition operators between the weighted Bergman space and the Zygmund space. Second, we study the characterizations of Schatten p-class Hankel operator on the harmonic Bergman spaces.Let D be the unit disk in the complex plane C, and let H(D) be the class of all holomorphic functions on D. For 0＜p＜∞,α＞-1, the weighted Bergman space A_α~p is defined byHere dA denotes the normalized Lebesgue area measure on D.The Zygmund space￡is defined bywhereIt is well known that It is easy to see that (?) is a Banach space under the norm (?), whereThe little Zygmund space of ED, denoted by (?), is the closed subspace of (?) consisting of functions f withGivenφan holomorphic self-map of D and g∈H(B), the product of extended Ces(?)ro and composition operator T_gC_φis defined byThis is a generalization of the extended Ces(?)ro operator. Ifφ(z) = z, then T_gC_φis just the extended Ces(?)ro operatorThe extended Ces(?)ro operator is significant in the operator theory of holomorphic functionspaces. Therefore, it is necessary to study this operator T_gC_φon the holomorphic function spaces. We characterize the boundedness and compactness of the operator T_gC_φbetween the weighted Bergman space and the Zygmund space (little Zygmund space). According to this, we enlarge the research field of operators, and give a deeper clarification of this operator.LetΩbe a bounded smooth domain in R~n(n≥2) and let V be the Lebesgue measure onΩ. Denote by L~2(Ω) the set of all measurable functions f onΩsuch that The harmonic Bergman space L_h~2(Ω) is the set of all harmonic functions in L~2(Ω).For f∈L~2(Ω), the multiplication operator M_f is defined by M_f(g)=fg. Let Q be the orthogonal projection from L~2(Ω) onto L_h~2(Ω), the Hankel operator H_f is defined on L~2(Ω) byWe discuss the characterizations of the Schatten p-class Hankel operator on the harmonicBergman spaces L_h~2(Ω), and obtain the necessary and sufficient conditions that H_f to belong to S_p(2≤p＜∞). Our work extends the results in references.