The Douglas question on the Bergman and Fock spaces

TL;DR

This paper investigates the invertibility of Toeplitz operators on Bergman and Fock spaces, providing necessary and sufficient conditions, and demonstrates that the Douglas question has negative answers in general.

Contribution

It offers new criteria for invertibility of Toeplitz operators on these spaces and shows limitations of Berezin transform-based characterizations.

Findings

Invertibility characterized by Berezin transform and Carleson conditions on Bergman space.

Invertibility on Fock space linked to reverse Carleson measures, not solely Berezin transform.

Counterexamples show the Douglas question has negative answers in general cases.

Abstract

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.