Weighted Composition Operators from the Bloch Space and the Analytic Besov Spaces into the Zygmund Space

Flavia Colonna,Songxiao Li

doi:10.1155/2013/154029

Abstract

We provide several characterizations of the bounded and the compact weighted composition operators from the Bloch space and the analytic Besov spaces (with ) into the Zygmund space . As a special case, we show that the bounded (resp., compact) composition operators from , , and to coincide. In addition, the boundedness and the compactness of the composition operator can be characterized in terms of the boundedness (resp., convergence to 0, under the boundedness assumption of the operator) of the Zygmund norm of the powers of the symbol.

Highlights

Let D be the open unit disk in the complex plane C and H(D) the space of analytic functions on D
We provide several characterizations of the bounded and the compact weighted composition operators from the Bloch space B and the analytic Besov spaces Bp into the Zygmund space Z
We show that there are no nontrivial bounded multiplication operators from the Bloch space to the Zygmund space, and that a composition operator from B or Bp into Z is bounded if and only if it is bounded as an operator acting on the space H∞ of bounded analytic functions on D under the supremum norm

Summary

Introduction

Let D be the open unit disk in the complex plane C and H(D) the space of analytic functions on D. Besides giving characterizations of the bounded and the compact operator uCφ from the Besov and the Bloch spaces into Z in terms of function theoretic conditions on the symbols u and φ, we provide boundedness and the compactness criteria for the operator uCφ, in terms of the norms in Z of functions in the range of the operator belonging to certain one-parameter families. For both boundedness and compactness, one of these characterizations involves the sequence {‖uφk‖Z}. We found this phenomenon quite surprising due to the size of the spaces Bp and B in comparison to that of the Zygmund space

Boundedness

Compactness

Multiplication and the Composition Operators