Some Problems Of Operators On Zygmund Type Space

We mainly study generalized composition operator, product of differential operator and weighted composition operator, composition operator. We use op-erator definitions, some properties of the norm and select test function, obtain necessary and sufficient conditions that these operators are bounded operators or compact operators in different holomorphic function spaces. This thesis consists of4chapters:In the first chapter, we discuss weighted composition operator, generalized composition operator, product of differential operator and weighted composition operator of the research background and give some basic concepts and notations. At last, we show the significance of research work.In the second chapter, the boundedness and compactness of generalized com-position operators on Zygmund type spaces in the unit disk are investigated in this paper. Further, we can get the boundedness and compactness of composition op-erators on Zygmund type spaces in the unit disk.In the third chapter, we study the boundedness and compactness of prod-ucts of differential operator and weighted composition operator on Zygmund type space in the unit disk, to find the conditions that the holomorphic function u and holomorphic self-map φ satisfied.In the fourth chapter, we use a new test function studying composition op- erator from Zygmund space to Bloch type space in the unit ball and obtain the necessary and sufficient conditions for bounded operator.