Generalized composition operators from Bloch type spaces toQKtype spaces

Fang Zhang,Yongmin Liu

doi:10.1155/2010/296570

Abstract

This paper characterizes the boundedness and compactness of the generalized composition operator(Cφgf)(z)=∫0zf'(φ(ξ))g(ξ)dξfrom Bloch type spaces toQKtype spaces.

Highlights

Let φ be an analytic self-map of the unit disk D
For g ∈ H(D), the class of all analytic functions on D, we define a linear operator as follows z (Cφg f )(z) = f (φ(ξ))g(ξ)dξ, f ∈ H(D)
When g = φ, we see that this operator is essentially composition operator Cφ which is defined by Cφf = f ◦ φ

Summary

Introduction

Let φ be an analytic self-map of the unit disk D. One of the critical problems on composition operators is to relate function theoretic properties of φ to operator theoretic properties of the restriction of Cφ to various Banach spaces of analytic functions. Composition operators from Bloch type spaces operators on Bα, Qp, F (p, q, s) and Qk have been studied by some authors (see, for example, [3,6,7,10,12] and references therein). The purpose of this paper is to study the boundedness and compactness of the generalized composition operators from Bloch type spaces to QK type spaces by the K− Carleson measure, which can be viewed as a development of the study on spaces QK and F (p, q, s). Throughout this paper, C always denote positive constants and may be different at different occurrences

Preliminaries and lemmas

Main results