Invertibility of Bergman Toeplitz operators
Mo Javed, Amit Maji
TL;DR
The paper addresses when Bergman Toeplitz operators T_phi on L^2_a(D) are invertible by linking invertibility to the positivity of the Berezin transform phi_tilde. For symbols phi = c g + d conjugate(g) with g in H∞(D), it proves T_phi is invertible iff inf_{z in D} |phi_tilde(z)| > 0 (equivalently inf_{z in D} |phi(z)| > 0), and it does so by a case split: analytic/coanalytic special cases and the mixed harmonic case analyzed through s = c/d, with |s| = 1 (normal) and |s| ≠ 1 (hyponormal) scenarios. The results extend prior work on Bergman Toeplitz invertibility, clarifying when the Berezin transform criterion suffices for harmonic symbols, and provide a concrete example illustrating the theory. This contributes to a deeper understanding of Douglas-type questions in Bergman space operator theory and offers a practical invertibility criterion for a broad class of harmonic symbols.
Abstract
In this paper, we establish the invertibility of the Berezin transform of the symbol as a necessary and sufficient condition for the invertibility of the Toeplitz operator on the Bergman space $L^2_a(\mathbb{D})$. More precisely, if $φ = c g + d \bar{g}$, where $c,d\in\mathbb{C}$ and $g\in H^{\infty}(\mathbb{D})$, the space of all bounded analytic functions, then $T_φ$ is invertible on $L^2_a(\mathbb{D})$ if and only if $\inf\limits_{z\in \mathbb{D}}\left|\widetilde{\,φ}(z)\right|=\inf\limits_{z\in \mathbb{D}}|φ(z)|>0$, where $\widetilde{\,φ}$ is the Berezin transform of $φ$.