Bergman空间的再生核与Toeplitz算子的特征向量

The Reproducing Kernel of Bergman Space and the Eigenvectors of Toeplitz Operator

在Bergman空间中,对任意φ∈¯¯H∞,众所周知TφKz=φ(z)Kz,即Kz是Tφ的属于φ(z)的特征向量,其中Kz是Bergman空间的再生核.反过来,φ是有界调和函数,若存在z∈D(或者对每一个z∈D)使得Kz是Tφ的特征向量,是否必有φ∈¯¯H∞?针对这些问题,该文给出了以再生核Kz为特征向量的具有有界调和符号Toeplitz算子的完全刻画,而且还给出了以所有的φ(z)(z∈D)为特征值的具有有界调和符号Toeplitz算子的部分刻画.

In the Bergman space,it is well-known that TφKz=φ(z)Kz forφ∈¯¯H∞,that is,Kz is the eigenvector of Tφcorresponding the eigenvalueφ(z),where Kz is the reproducing kernel of Bergman space.Conversely,ifφis a bounded harmonic function and if there is z∈D(or for every z∈D),Kz is a eigenvector of Tφ,whether there must beφ∈¯¯H∞?In view of the above questions,in this paper we give a complete characterization of the Toeplitz operator with the bounded harmonic symbol which have the reproducing kernels Kz as their eigenvectors.Moreover,we partially describe the Toeplitz operators with the bounded harmonic symbol whose eigenvalues are allφ(z)(z∈D).