Multipliers on the Dirichlet Space for the Annulus
Received:November 21, 2017  Revised:January 13, 2018
Key Word: annulus   multiplier   spectra   essential spectra  
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11501136), Featured Innovation Project of Guangdong Province (Grant No.2016KTSCX105) and Youth Project of Guangzhou City (Grant No.1201630152).
Author NameAffiliation
Zelong CAO Zhixin High School, Guangdong 510080, P. R. China 
Junlin LIU Zhixin High School, Guangdong 510080, P. R. China 
Li HE Department of Mathematics, Guangzhou University, Guangdong 510006, P. R. China 
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      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.
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