The Study Of Properties Of Composition Operators And Invariant Subspaces

Combining the theory of complex functions with several variables, and operator theory, as well as the theory of invariant subspace, in this paper we mainly discuss the boundedness, compactness, essential norms and differences of composition operators,furthermore, we talk about a property concerning the invariant subspace. The whole paper includes six chapters:In the first chapter, we present the background and present situation of the study,and then lead to the problems to be investigated.In the second chapter, basic concepts and relative properties are introduced.In the third chapter, the object of study is high order differential type composition operators, we give the necessary and sufficient conditions of the n-differentiation com-position operators from weighted Banach spaces of holomorphic functions to weight-ed Bloch spaces, and the composition operators followed by n-differentiation acting from QK(p,q) spaces to Bloch-type spaces in the unit disk, respectively. Although,the method used here is classical, but the subject considered in previous results mainly focus on composition operators, weighted composition operators, integral-type composi-tion operators and so on, however, the innovation of this section is that we are discussing the high order differential type composition operators, and then we need special skills to deal with the problem.In the forth chapter, we go much further toward the differences of differentia-tion type composition operators, discussing on the differences of composition operators followed by differentiation between weighted Banach spaces of holomorphic functions,and the differences of weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces, all the spaces considered here is based on the unit disc.The fifth chapter is devoted to the new characterization of boundedness and es-sential norm of integral-type composition operators acting on Bloch-type spaces in the unit disc and the unit ball, it should be noted that there is a great difference between the method used here and the classical one used above.Finally, in the sixth chapter, further sublimation about the study of operators, we consider the maximal almost-invariant subspaces and maximal hyperinvariant subspaces respect to different, types of operators.