Multipliers on the Dirichlet Space for the Annulus

摘要

Multipliers on the classic Dirichlet space of the unit disk are much more complex than those on the Hardy space and the Bergman space,many basic problems have not been solved,such as the boundedness,which is still an open problem.The annulus,as a kind of typical complex connected domain,has more complicated function structure.This paper focuses on discussing the invertibility and Fredholmness of multipliers on the Dirichlet space of the annulus.The spectra and essential spectra of multipliers with Laurent polynomials symbols are calculated.In addition,we anwser a problem proposed by Guangfu CAO and Li HE on spectrum and essential spectrum for general multipliers.