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