Abstract
For any analytic self-map \({\varphi}\) of {z : |z| < 1} we give four separate conditions, each of which is necessary and sufficient for the composition operator \({C_{\varphi}}\) to be closed-range on the Bloch space \({\mathcal{B}}\) . Among these conditions are some that appear in the literature, where we provide new proofs. We further show that if \({C_{\varphi}}\) is closed-range on the Bergman space \({\mathbb{A}^2}\) , then it is closed-range on \({\mathcal{B}}\) , but that the converse of this fails with a vengeance. Our analysis involves an extension of the Julia-Carathéodory Theorem.
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An erratum to this article can be found online at http://dx.doi.org/10.1007/s00020-013-2043-7.
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Akeroyd, J.R., Ghatage, P.G. & Tjani, M. Closed-Range Composition Operators on \({\mathbb{A}^2}\) and the Bloch Space. Integr. Equ. Oper. Theory 68, 503–517 (2010). https://doi.org/10.1007/s00020-010-1806-7
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DOI: https://doi.org/10.1007/s00020-010-1806-7