Composition Operators On Reproducing Analytic Hilbert Spaces

Suppose analytic function g(z) = (?)a_nZⁿ(a_n ≥ 0,n = 0,1,2… )with theradius of convergence R², H_g²(D_r) denotes the general analytic function spaces which is produced by g(z). We call H_g²(D_R) the reproducing analytic Hilbert spaces. They contain many classical spaces of analytic function such as: Hardy space H²(D), Bergman space L_a²(D), Dirichlet space D(D), Fock space L_a²(C) and the respective weighted spaces . This paper deals with the generators of reproducing analytic Hilbert spaces H_g²(C), the Hilbert-Schmidt composition operators on H_g²(D), the compact differences of composition operators on reproducing analytic Hilbert spaces H_g²(D), and some topological properties of C(H_g²(D_r)).Chapter one is concerned with the basic structure of the reproducing analytic Hilbert spaces H_g²(D_R).In chapter two, we study the generators of analytic Hilbert spaces E²{γ) which is a case of H_g²(D_r) with some special analytic function g(z) .The main result is that if α,β,∈C, |α| < (?) = (?) nγ_n/γ_n-1, then e^αz+β is a generator of E²(γ)Chapter three is the main part of this paper. When H_g² (D) is a subnormal space, the Hilbert-Schmidt composition operators on H_g²(D) are characterized. In this chapter, we survey the compactness of the differences of composition operators acting on H_g²(D_R). A necessary condition of compact difference is given for composition operators acting on H_g²(D_R). Specially, suppose g(z) = (1/(1-z))^β(β>0) and H_g²(D) is a Carleson subnormal space, p{z) =(?) we nave: (C_?-C_ψ) is compact if and only if(?)...