Toeplitz operators with BMO symbols and the Berezin transform (Q1415144)

Let \(L^2_a(D)\) denote the Bergman space which is the subspace in \(L^2(D)\) consisting of analytic functions on the unit disk \(D\) and let \(T_f g=P(fg)\) be the Toeplitz operator with symbol \(f\) where \(P:L^2(D)\to L^2_a(D)\) is the Bergman projection. For \(f\in L^1 (D)\), denote by \(\widetilde f\) its Berezin transform, that is, \(\widetilde f (z)=\int_D f(\omega)| k_z(\omega)| ^2\,dm(\omega)\), where the normalized kernel function \(k_z\) has the form \(k_z(\omega)=\frac{1-| z| ^2}{(1-\overline z \omega)^2}\). Finally, \(f\in L^1 (D)\) belongs to \(\text{BMO}^p (D)\), \(p\geq 1\), if \(\sup \| f\circ \psi_z -\widetilde f (z)\| _p<\infty\), where \(\psi_z\) denotes the disk automorphism. The author studies the boundedness and compactness of the Toeplitz operator on the Bergman space with a \(\text{BMO}^1\) symbol in terms of its Berezin tranform. In particular, the operator \(T_f\) is bounded if and only if \(\widetilde f\) is bounded. The main result contains the following Theorem. Let \(f\) belong to \(\text{BMO}^1(D)\). Then \(\widetilde f \to 0\) as \(z\to \partial D\) implies that \(T_f\) is compact on \(L_a^2(D)\).