Toeplitz operators with BMO symbols of several complex variables (Q1758015)

Let \(\mathbb{B}\) be the open unit ball in \(\mathbb{C}^n\), \(d\nu\) its normalized Lebesgue measure, and \(BMO^1(\mathbb{B})\) the space of functions \(\psi\in L^1(\mathbb{B})\) satisfying \[ \sup_{z\in\mathbb{B}} \|\psi(\varphi_z)-\tilde{\psi}(z)\|_{L^1(d\nu)}<\infty, \] where \(\varphi_z\) is the involutive automorphism of \(\mathbb{B}\) which interchanges \(0\) and \(z\) and \(\tilde {\psi}\) denotes the Berezin transform of \(\psi\) defined by \[ \tilde \psi(z)=\int_{\mathbb{B}} \psi(\varphi_z(w))\,d\nu(w). \] The authors prove that, if \(\psi\in BMO^1\), then the Toeplitz operator \[ T_\psi(f)(z)=\int_{\mathbb{B}} \frac{f(w)\psi(w)}{(1-z\overline w)^{n+1}}\,d\nu(w) \] is compact on the Bergman space \(A^2(\mathbb{B})\) if and only if \(\tilde \psi(z)\to 0\) as \(|z|\to 1\), extending a result in [\textit{N. Zorboska}, Int. J. Math. Math. Sci. 2003, No. 46, 2929--2945 (2003; Zbl 1042.47022)] for the one dimensional case. A generalization of this result for weighted Toeplitz operators on weighted Bergman spaces on \(\mathbb{B}\) was proved simultaneously in [\textit{K. Zhang} et al., Acta Math. Sin., Engl. Ser. 27, No. 11, 2129--2142 (2011; Zbl 1243.47065)].