Closed-Range Composition Operators on {\mathbb{A}^2} and the Bloch Space |
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| Authors: | John R Akeroyd Pratibha G Ghatage Maria Tjani |
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| Institution: | 1. Department of Mathematics, University of Arkansas, Fayetteville, AR, 72701, USA
2. Department of Mathematics, Cleveland State University, Cleveland, OH, 44115, USA
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| Abstract: | For any analytic self-map j{\varphi} of {z : |z| < 1} we give four separate conditions, each of which is necessary and sufficient for the composition operator Cj{C_{\varphi}} to be closed-range on the Bloch space B{\mathcal{B}} . Among these conditions are some that appear in the literature, where we provide new proofs. We further show that if Cj{C_{\varphi}} is closed-range on the Bergman space
\mathbbA2{\mathbb{A}^2} , then it is closed-range on B{\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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