Composition Operators And Extended Cesàro Operators Related To Holomorphic Q_k Spaces

We study the boundedness and compactness of products of composition,differentiation and multiplication operators and extended Cesàro operator related holomorphic QK spaces in this paper. We apply the properties of operators, norm,function space and select test function to obtain the sufficient and necessary conditions that these operators are bounded and compact in some holomorphic function spaces. The main contents of this paper are as follows:1. Introduce the background of research and main contents of this paper.2. Introduce some important concept and lemmas which are the foundations and tools for the later research.3. First introduce some necessary properties of Bloch space and Q, space. In this section, we mainly study the boundedness and compactness of the products of the composition and differentiation operators from Bloch space to QK space. We overcome the problem of integration from Bloch spaces into QK spaces and obtain sufficient and necessary conditions for the operator C?D :B ?QK in the unit disk to be bounded and compact.4. First introduce some necessary properties of Zygmund space and the weighted-type space. We obtain sufficient and necessary conditions for Stevic -Sharma operators from Zygmund spaces into weighted-type spaces in the unit disk to be bounded and compact. Further, we can get sufficient and necessary conditions for weighted composition operator W?,?:Z?H?? and products of differentiation and weighted composition operator DW?,?:Z?H?? in the unit disk to be bounded and compact. This section generalize the results of composition operator.5. In this chapter, we overcome the problem of integration from a-Bloch spaces into QK spaces and characterize the boundedness and compactness of extended Cesàro operators from ?-Bloch spaces into QK spaces in the unit disk,which makes contribution to the exploration in this field.