A Note on Closed-Range Composition Operators

摘要

Let be an analytic mapping from the unit disk to itself. In the context of Banach spaces of analytic functions on , we address the question: Does closed-rangedness of the composition operator on one space confer closed-rangedness of on a smaller space? We find that the answer is quite often in the affirmative and provide conditions that guarantee this. We also show that, in the absence of these conditions, the answer is not always in the affirmative.