Some Kinds Of Linear Operators On Certain Holomorphic Function Spaces

In this thesis we deal with some kinds of linear operators between certain spaces of holomorphic functions. We denote by H(D) the class of all holomorphic functions on the unit disc D in the complex plane C. Given φ a holomorphic self-map of D, a linear operator C_φ on H(D), called a composition operator, is defined by C_φf = fοφwith the symbol φ. At present, the theory of the composition operator is one of the hot points in research field of function theory. We characterize the boundedness and compactness of the operator C_φ between the weighted Bergman spaces and the Bloch-type spaces as well as the little Bloch-type spaces, which gives a deeper clarification of the composition operator. Given u a fixed holomorphic function on D and φ a holomorphic self-map of D, the weighted composition operator uC_φ can. be defined byuC_φf = ufοφwhere f ∈ H(D). It is obvious that this operator is the product of the multiplication operator and the composition operator, which also can be regarded as a generalization of two kinds of the operators. As we all know those operators are very significant in the linear operator theory of holomorphic function spaces, therefore it is very necessary to study this operator uC_φ. We discuss the boundedness and compactness of uC_φ between the Bers-type spaces (or the little Bers-type spaces) and the weighted Bergman spaces, and give some estimates for the norm of uC_φ between those spaces. At the same time, we study the operator uC_φ between H~∞ space and the weighted Bergman spaces, and obtain some sufficient and necessary conditions for uC_φ to be bounded or...