The Douglas question on the Bergman and Fock spaces

Let μ be a positive Borel measure and T_μ be the bounded Toeplitz operator induced by μ on the Bergman or Fock space. In this paper, we mainly investigate the invertibility of the Toeplitz operator T_μ 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_μ in terms of the Berezin transform of μ and the reverse Carleson condition in two classical cases: (1) μ is absolutely continuous with respect to the normalized area measure on the open unit disk D; (2) μ is the pull-back measure of the normalized area measure under an analytic self-mapping of 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_μ is invertible if and only if μ is a reverse Carleson measure, but the invertibility of T_μ is not completely determined by the invertibility of the Berezin transform of μ. 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.