Copyright © 2011 Nina Zorboska. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We present an overview of the known results describing the isometric and closed-range composition operators on different types of holomorphic function spaces. We add new results and give a complete characterization of the isometric univalently induced composition operators acting between Bloch-type spaces. We also add few results on the closed-range determination of composition operators on Bloch-type spaces and present the problems that are still open.