The Weighted Differentiation Composition Operators On Different Spaces

Weighted composition differentiation operation is one of the important content in operator theory,which reflects the relationship between properties of differentiation operators and properties of the defined functions,and builds the relationship between operator theory and function theory.In this paper,we mainly use the theory of function space to study the properties of the weighted composition differential operator,on the space of analytic function,fixed on the unit open disk.For example Bers space,from Bloch space into weighted-type space.There are three parts in this paper.Firstly,we study the compactness and boundedness of product of composition and differentiation operators of uDC_? on the Bloch space on the unit disk,we obtain a sufficient and a necessary condition for uDC_? operators are compact on the Bloch space.Then,we consider the compactness and boundedness of product of weighted differentiation and composition operators of uDC_? on the Bers-type space on the unit disk,we obtain this condition turns out to be necessary and sufficient for uDC_? operator is compact and bounded on the Bers-type space.