Generalized Composition Operators on Zygmund-Orlicz Type Spaces and Bloch-Orlicz Type Spaces

Congli Yang,Pengcheng Wu,Fangwei Chen

doi:10.1155/2014/549370

Abstract

The boundedness and compactness of generalized composition operators on Zygmund-Orlicz type spaces and Bloch-Orlicz type spaces are established in this paper.

Highlights

Let D be a unit disk in complex plane C, and let H(D) be the space of all holomorphic functions on D with the topology of uniform convergence on compact subsets of D
In [6], the boundedness and compactness of the generalized composition operator on Zygmund space and Bloch-type spaces are characterized by Li and Stevic
Since Cφgf(0) = 0, it follows that the generalized composition operator Cφg : Z → Bφ is bounded

Summary

Introduction

Let D be a unit disk in complex plane C, and let H(D) be the space of all holomorphic functions on D with the topology of uniform convergence on compact subsets of D. Inspired by the way Bloch-Orlicz spaces were defined (see [8, 14]), we define the Zygmund-Orlicz space, which is denoted by Zφ, as the class of all analytic functions f in D such that sup z∈D (20). In [6], the boundedness and compactness of the generalized composition operator on Zygmund space and Bloch-type spaces are characterized by Li and Stevic. We are devoted to investigating the boundedness and compactness of generalized composition operators between Zygmound-Orlicz type spaces and BlochOrlicz type spaces. Since Cφgf(0) = 0, it follows that the generalized composition operator Cφg : Z → Bφ is bounded.

So we have

We have

Therefore lim