The Douglas question on the Bergman and Fock spaces

Let $\mu$ be a positive Borel measure and $T_\mu$ be the bounded Toeplitz operator induced by $\mu$ on the Bergman or Fock space. In this paper, we mainly investigate the invertibility of the Toeplitz operator $T_\mu$ and the Douglas question on the Bergman and Fock spaces. In the Bergman-space setting, we obtain several necessary and sufficient conditions for the invertibility of $T_\mu$ in terms of the Berezin transform of $\mu$ and the reverse Carleson condition in two classical cases: (1) $\mu$ is absolutely continuous with respect to the normalized area measure on the open unit disk $\mathbb D$; (2) $\mu$ is the pull-back measure of the normalized area measure under an analytic self-mapping of $\mathbb D$. Nonetheless, we show that there exists a Carleson measure for the Bergman space such that its Berezin transform is bounded below but the corresponding Toeplitz operator is not invertible. On the Fock space, we show that $T_\mu$ is invertible if and only if $\mu$ is a reverse Carleson measure, but the invertibility of $T_\mu$ is not completely determined by the invertibility of the Berezin transform of $\mu$. These suggest that the answers to the Douglas question for Toeplitz operators induced by positive measures on the Bergman and Fock spaces are both negative in general cases.