Abstract:

In the Bergman space, it is well-known that $T_{\varphi}K_{z}=\varphi(z)K_{z}$ for $\varphi\in \overline{H^{\infty}}$, that is, $K_{z}$ is the eigenvector of $T_{\varphi}$ corresponding the eigenvalue $\varphi(z)$, where $K_{z}$ is the reproducing kernel of Bergman space. Conversely, if $\varphi$ is a bounded harmonic function and if there is $z\in \mathbb{D}$ (or for every $z\in\mathbb{D}$), $K_{z}$ is a eigenvector of $T_{\varphi}$, whether there must be $\varphi\in \overline{H^{\infty}}$ ? 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 $K_{z}$ as their eigenvectors. Moreover, we partially describe the Toeplitz operators with the bounded harmonic symbol whose eigenvalues are all $\varphi(z) (z\in \mathbb{D})$.

Key words: Bergman space, Reproducing kernel, Toeplitz operator, Eigenvectors