Topological And Algebraic Properties Of Composition Operators

LetΩbe a domain in Euclidean complex space CN,φbe an analytic self map ofΩ, and u be a holomorphic function onΩ. If f is an element of an analytic functions space, a linear operator named composition operator which is induced byφis defined by Cφf= f oφ, and a weighted composition operator is defined by uCφ(f)=u·f○φ.In this thesis, topological connectness and compact linear combination of derivative weighted composition operators on classic Bergman space are firstly investigated. Then topological structure and linear combination of non weighted composition operators on Dirichlet space are studied as corollaries.Secondly, on the unit ball algebra and bounded analytic function space on the unit ball in CN, we characterized the compact linear combination and difference of weighted composition operators.Finally, we focused on the (compact) intertwining relation for integral type and composition operators. On several function spaces in the unit disk, H∞, Bloch and Bergman space etc., we discussed essential commutativity between that integral type and composition operators.