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We show that there exist non-compact composition operators in the connected component of the compact ones on the classical Hardy space H 2. This answers a question posed by Shapiro and Sundberg in 1990. We also establish an improved version of a theorem of MacCluer, giving a lower bound for the essential norm of a difference of composition operators in… (More)

A remarkable result by Denjoy and Wolff states that every analytic self-map ϕ of the open unit disc D of the complex plane, except an elliptic automorphism, has an attractive fixed point to which the sequence of iterates {ϕn} n1 converges uniformly on compact sets: if there is no fixed point in D, then there is a unique boundary fixed point that does the… (More)

The norm of a bounded composition operator induced by a disc automorphism is estimated on weighted Hardy spaces H 2 (β) in which the classical Hardy space is continuously embedded. The estimate obtained is accurate in the sense that it provides the exact norm for particular instances of the sequence β. As a by-product of our results, an estimate for the… (More)

The boundedness and compactness of weighted composition operators on the Hardy space H 2 of the unit disc is analysed. Particular reference is made to the case when the self-map of the disc is an inner function. Schatten-class membership is also considered; as a result, stronger forms of the two main results of a recent paper of Gunatillake are derived.… (More)

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