The compact of the composition operator cλ

Abstract EN

Let U denote the unit ball in the complex plane, the Hardy space H2 is the set of functions f(z)=∑_(n=0)^∞▒〖f^(n) 〗 zn holomorphic on U such that ∑_(n=0)^∞▒〖|f^(n)| 〗 < ∞ with f^(n) denotes the Taylor coefficient of f .

Let  be a holomorphic self-map of U, the composition operator induced by C is defined on H2 by the equation C = f °  (f ∈H2) We have studied the composition operator induced by the bijective map  and discussed the adjoint of the composition operator .We have look also at some known properties of composition operator and tried to get the analogue properties in order to show how the results are changed by changing the map  in U.

In order to make the work accessible to the reader , we have included some known results with the details of the proofs for some cases and proved some results.