On the Compactness of the Composition Operator  C

Abstract

Let U denote the unit ball in the complex plane, the Hardy space 2 H is the set of functions

ï‚¥


n 0
^ n f (z) f (n) z holomorphic on U such that   
ï‚¥

2
n 0
^ f (n) with f (n) ^ denotes then the
Taylor coefficient of f .
Let ï¹ be a holomorphic self-map of U, the composition operator ï¹ C induced by ï¹ is defined
on 2 H by the equation
C f f (f H ) 2  ï¹ ïƒŽ ï¹ ï¯
In this paper we have studied the compactness of the composition operator induced by the
bijective map ï³ and discussed the adjoint the compactness of the composition operator of the map ï³
.We give some theorems on compactness of the composition operators. We have look also at some
known properties on composition operators and tried to see the analogue properties in order to show
how the results are changed by changing the map ï¹ 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 proofs for the properties .