Operators on the orthogonal complement of the Dirichlet space (Ⅱ)  

作　　者：YU Tao 

机构地区：[1]Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

出　　处：《Science China Mathematics》2011年第9期2005-2012,共8页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China (Grant Nos.10971195 and 10771064);Natural Science Foundation of Zhejiang Province (Grant Nos. Y6090689 and Y6110260);Zhejiang Innovation Project (Grant No. T200905)

摘　　要：In this paper we first prove that a dual Hankel operator Rφ on the orthogonal complement of the Dirichlet space is compact for φ ∈ W^1,∞（D）, and then that a semicommutator of two Toeplitz operators on the Dirichlet space or two dual Toeplitz operators on the orthogonal complement of the Dirichlet space in Sobolev space is compact. We also prove that a dual Hankel operator Re with φ ∈ W^1,∞（D） is of finite rank if and only if Be is orthogonal to the Dirichlet space for some finite Blaschke product B, and give a sufficient and necessary condition for the semicommutator of two dual Toeplitz operators to be of finite rank.In this paper we first prove that a dual Hankel operator R φ on the orthogonal complement of the Dirichlet space is compact for φ∈ W 1,∞(D),and then that a semicommutator of two Toeplitz operators on the Dirichlet space or two dual Toeplitz operators on the orthogonal complement of the Dirichlet space in Sobolev space is compact.We also prove that a dual Hankel operator R φ with φ∈ W 1,∞(D) is of finite rank if and only if B φ is orthogonal to the Dirichlet space for some finite Blaschke product B,and give a sufficient and necessary condition for the semicommutator of two dual Toeplitz operators to be of finite rank.

关 键 词：Sobolev space Dirichlet space dual Toeplitz operator dual Hankel operator Toeplitz operator 

分 类 号：O177.1[理学—数学] O17-4[理学—基础数学]