Closed-Range Composition Operators on mathbbA2{mathbb{A}^2} and the Bloch Space

摘要

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 C_j{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 C_j{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.