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Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Composition operators on H^2 . | J. OPERATOR THEORY 9 1983 77 106 Copyright by INCREST 1983 COMPOSITION OPERATORS ON H2 CARL c. COWEN 1. INTRODUCTION For p analytic in the unit disk D such that p D D the composition operator C p is the operator on the Hardy space II2 of the unit disk given by Cọf f o p for all f in IT2. Ryff 24 showed that Cạ is always a bounded operator. Several authors have found that the properties of Cq depend to a great extent on the behaviour of p near its fixed points. We will say a point b in D is a fixed point of p if lim ep rb b. We will write p b for lim p rb the limit obviously exists r- l if h 1 and if ố 1 the theorem of Julia Carathéodory and Wolff 20 page 57 shows that this limit exists and 0 p b oo. Although it is not a priori evident that p has fixed points it has at least one. Denjoy-Wolff Theorem 11 29 1 . If p not an elliptic Mobius transformation ofD onto D is analytic in D with p D c D then there is a unique fixed point a of p with a 1 such that p a l I- We will call the distinguished fixed point a the Denjoy-Wolff point of p and we reiterate that if a 1 then 0 p a 1 and if a 1 then 0 p a 1. The results of this paper the most important of which are noted below strengthen the observation that properties of Cv depend on the behaviour of p near its fixed points. The hypotheses of all the following theorems include the assumption that p is analytic in D with p D D and that a denotes the Denjoy-Wolff point of p but we omit this statement for brevity. Theorem 2.1. If a 1 the spectral radius of Cọ is 1. If fl 1 the spectral radius of Cự is p d lli. If T is any operator the essential norm of T is IIT lle inf ị T AII A is a compact operator and the essential spectrum of T is off p p T is not a 78 CARL c. COWEN Fredholm operator . The following result is based on a new estimate of the radial maximal function due to B. J. Davis Theorem 2.2 . Theorem 2.4. If p is continuous on D then ịịCJ e 2M1 2 where M - max Ịợ ei0 j 1 21 1 . Under more restrictive hypotheses tats is .
