Commuting Toeplitz operators on the Bergman space of an annulus (Q1923903)

Let \(\Omega= \{z\in\mathbb{C}: R<| z|<1\}\), \(0< R<1\), be an annulus in the complex plane, and \(L^2_a(\Omega)\) be the Bergman space of the analytic functions on \(\Omega\) that are square integrable on \(\Omega\) with respect to the area measure \(dA\). If \(\varphi\in L^\infty(\Omega)\), the Toeplitz operator \(T_\varphi: L^2_a(\Omega)\to L^2_a(\Omega)\) is defined by \(T_\varphi(f)= P(\varphi f)\), where \(P: L^2(\Omega)\to L^2_a(\Omega)\) is the orthogonal projection. In this paper, the author proves that: Suppose that \(\varphi= f_1+\overline f_2\), \(\phi= g_1+\overline g_2\) are bounded harmonic functions on \(\Omega\), then \(T_\varphi T_\phi= T_\phi T_\varphi\) if and only if (1) \(\varphi\) and \(\phi\) are both analytic on \(\Omega\); or (2) \(\overline\varphi\) and \(\overline\phi\) are both analytic on \(\Omega\); or (3) there exist constants \(a,b\in\mathbb{C}\), not both \(0\), such that \(a\varphi+ b\phi\) is constant on \(\Omega\). The proof is based on the Berezin transform \[ (Bu)(w)= \int_\Omega u(z) | k_w(z)|^2 dA(z),\quad w\in\Omega; \] where \(k_w\) is the normalized reproducing kernel of \(L^2_a(\Omega)\).