Isometric and Closed‐Range Composition Operators between Bloch‐Type Spaces

Nina Zorboska

doi:10.1155/2011/132541

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.

Highlights

A topic of interest in the paper is the description of isometric and, more generally, of closedrange composition operators on the Bloch-type spaces, in terms of the specific behaviour of the inducing function
The goal of the paper is to present an overview of the known results by emphasizing the intuitive idea and geometrical aspects of the corresponding conditions, to contribute to the classification with few new results and to list a number of open questions related to this topic
One of the earliest results on isometric composition operators, acting on spaces of functions analytic on the open unit disk, is Nordgren’s result 1 from 1968: if φ is inner, Cφ is an isometry on H2 if and only if φ 0 0

Summary

Introduction

A topic of interest in the paper is the description of isometric and, more generally, of closedrange composition operators on the Bloch-type spaces, in terms of the specific behaviour of the inducing function. In most of the cases, the general rule is that a composition operator is either isometric or has a closed range, whenever the image of the unit disc under the inducing function covers a significant in some sense part of . For general results and references on composition operators acting on various spaces of analytic functions, see, for example, 7, 8 .

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Results

Conclusion