Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.5, No.6, 2015 193 The Product of Finite Numbers of Automorphic Composition Operators on Hardy Space  2 Eiman H. Abood 1* Samira N. Kadhim 2* Sarah M. Khalil 3 1. University of Baghdad, College of Science, Department of Mathematics 2. University of Baghdad, College of Science, Department of Mathematics. 3. University of Baghdad, College of Science for women, Department of Mathematics. * E-mail of the corresponding author: eimanmath@yahoo.com 1* samira.naji@yahoo.com 2* Abstract Throughout this paper we study the properties of the composition operator C 1 p  o 2 p  o…o n p  induced by the composition of finite numbers of special automorphisms of U, i p  (z)  i i p z 1 pz   such that p i  U, i  1, 2, …, n, and discuss the relation between the product of finite numbers of automorphic composition operators on Hardy space  2 and some classes of operators. Keywords: Composition operator, Normal composition operator, Compact operator 1. Introduction Let U denote the unit ball in the complex plane, the Hardy space  2 is the collection of functions f(z) n n 0 ˆ f(n)z    , which holomorphic on U such that 2 n 0 ˆ |f(n)|     with ˆ f(n) denoting the n-th Taylor coefficient of f, and the norm of f is defined by: || f ||2  2 n 0 ˆ |f(n)|    . The particular importance of  2 is due to the fact that it is a Hilbert space. Let  be a holomorphic self-map of U, the composition operator C  induced by  is defined on  2 by the equation C  f  fo, for every f   2 , (see [9]). A conformal automorphism of U is a univalent holomorphic mapping of U onto itself. Each such map is a