Dirichlet空间D^H上Toeplitz算子

Toeplitz Operators on the Dirichlet Space D^H

给出了Sobolev空间W_H^(1,2)(D)的一个直和分解,研究了Dirichlet空间D^H上Toeplitz算子的(半)交换性,得到了一个充要条件。给出了交换算子T_φT_ψ-T_ψT_φ及半交换算子T_φT_ψ-T_(ψφ)有有限秩的一个充要条件,证明了Dirichlet空间D^H上Toeplitz算子为紧算子当且仅当它是零算子。

In this paper we give Sobolev space W1H^2）（ D） a direct sum decomposition. We study the（ semi-） commutativity of Toeplitz operators on the Dirichlet space DH,which gets a necessary and sufficient condition. Then a necessary and sufficient condition is obtained about that the commutator TφTψ- TψTφor the semi-commutator TφTψ-Tψφhas finite rank. Finally we prove a Toeplitz operator on the Dirichlet space D-His a compact operator if and only if it is a zero operator.