Composition Operators Between Some Banach Spaces Of Analytic Functions

In this thesis we deal with some kinds of linear operators between certain spaces of analytics functions. We denote by H(D) the class of allholomorphic functions on the unit disc D in the complex plane C. Given (?) an analytic self-map of D and a analytic function on D, we define alinear operator uC_? on H(D), called a weighted composition operator, byuC_?(z) = u(z)f((?)(z)), z∈DThis operator both generalizes a composition operator and a multiplication operator.In this thesis, we focus on three parts as follows: first we characterize the boundedness and compactness of the operator uC_? between the H^∞spaces and theα- Block type spaces as well as the littleα- Block -type spaces. Then we obtain the equation characterization for the operator C_?M_uto be bounded and compact on the little weighted Bloch spaces in the unit disk. Finally, necessary and sufficient conditions of compactness for a weighted composition operator uC_? are given by the locally uniformly convergence; atthe end of this thesis, a beautiful and interesting proof for a key equality appeared in Tran.A.M.S(1995) by Madigan and Matheson is given, the proof of the proposition was never published or reported before.