Acta mathematica scientia,Series A ›› 2014, Vol. 34 ›› Issue (4): 890-904.

• Articles • Previous Articles     Next Articles

Toeplitz Operator and Its Algebra on Dirichlet Space of the Annulus Domain

 CHEN Jian-Jun1, WANG Xiao-Feng1,2   

  1. 1.School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006;
    2.Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou 510006
  • Received:2012-11-14 Revised:2014-01-10 Online:2014-08-25 Published:2014-08-25
  • Supported by:

    广州市教育局高校科技计划项目(2012A018)资助.

Abstract:

In this paper, we have given two goals. Firstly we study a single Toeplitz operator Tφ with symbol φL∞,1, which is on the general Dirichlet space Dp (1<p<+∞) of the annulus domain, and further conclude that Tφ is compact if and only if its Berezin transform vanishes at the boundary of the annulus. Secondly we study the Toeplitz operator Tu with symbol $u\in uC1(M), which is on classical Dirichlet space D2, and further give a canonical decomposition S=TS+R for some S=∑mi=1nj=1Tuij in Toeplitz algebra T and some R in the commutator ideal CT.

Key words: Toeplitz operator, Berezin transform, Dirichlet space, Compact operator

CLC Number: 

  • 47B35
Trendmd