Invertibility of Toeplitz operators via Berezin transforms (Q2835244)

The main object under consideration in the present paper is the Toeplitz operator NEWLINE\[NEWLINE (T_\varphi f)(z)= \int_{\mathbb D} \frac{\varphi(w)f(w)}{(1-\bar w z)^2}\,\text{d}A(w) NEWLINE\]NEWLINE on the Bergman space \(L^2_a(\mathbb D)\). Here, \(\varphi\in L^\infty(\mathbb D)\) is the symbol of \(T_\varphi\) and \(\text{d}A\) is the normalized area measure on the unit disk. The problem the authors address concerns invertibility of \(T_\varphi\) in terms of the symbol. One of the main results states that, for nonnegative symbols, \(T_\varphi\) is invertible on \(L^2_a(\mathbb D)\) as long as NEWLINE\[NEWLINE \inf_{\mathbb D}\widetilde\varphi(z)>0, \quad \widetilde\varphi(z):=\int_{\mathbb D} \varphi(w)\,\frac{(1-| z| ^2)^2}{| 1-\bar w z| ^4}\,\text{d}A(w)NEWLINE\]NEWLINE is the Berezin transform of \(\varphi\). On the other hand, there is a symbol \(\varphi\in L^\infty(\mathbb D)\) so that \(| \widetilde\varphi(z)| \geq\delta>0\) on the unit disk, and the Toeplitz operator \(T_\varphi\) is irreversible on the Bergman space.