Spectrum of Compact Weighted Composition Operators on the Weighted Hardy Space in the Unit Ball

Ze-Hua Zhou,Cheng Yuan

doi:10.1155/2007/61313

Abstract

Let be the unit ball in the -dimensional complex space, for , a holomorphic function in , and , a holomorphic map from into itself, the weighted composition operator on the weighted Hardy space is given by , where . This paper discusses the spectrum of when it is compact on a certain class of weighted Hardy spaces and when the composition map has only one fixed point inside the unit ball.

Highlights

It is well known that the general principle that the spectrum structure of the composition operator Cφ is closely related to the fixed point behavior of the map φ is well illustrated by compact composition operators
About the spectrum of a compact operator in a weighted Hardy space defined in the disk or BN, we refer the reader to see [1], where Cowen and MacCluer proved a theorem of considerable generality, which will show that, essentially, all of the spaces of interest to us these eigenvalues are determined by the derivative of φ at the Denjoy-Wolff fixed point of φ
Weighting a composition operator as a generalization of a multiplication operator and a composition operator, recently, Gunatillake in [2] obtained some results for the spectrum of weighted composition operators on the weighted Hardy spaces of the unit disk

Summary

Recommended by Andras Ronto

This paper discusses the spectrum of Cψ,φ when it is compact on a certain class of weighted Hardy spaces and when the composition map φ has only one fixed point inside the unit ball

Introduction

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