The Douglas question on the Bergman and Fock spaces

Abstract

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 $\mathbb D$; (2) $μ$ 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 $T_μ$ 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.

Theorems & Definitions (34)

• Lemma 2.1
• Definition 2.2
• Definition 2.3
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Theorem 3.4
• proof
• Lemma 3.5
• Theorem 3.6