Schatten class Toeplitz operators on the Bergman space
Summary: We show that if the Toeplitz operator \(T_\varphi\) on the Bergman space \(La^2(\mathbb D)\) belongs to the Schatten class \(S^p\), \(1\leq p<\infty\), then \(\widetilde{\varphi}\in L^p(\mathbb D,d\lambda)\), where \(\widetilde{\varphi}\) is the Berezin transform of \(\varphi\), \(d\lambda(z)= dA(z)/(1-|z|^2)^2\), and \(dA(z)\) is the normalized area measure on the open unit disk \(\mathbb D\). Further, if \(\varphi\in L^p(\mathbb D,d\lambda)\), then \(\widetilde{\varphi}\in L^p(\mathbb D,d\lambda)\) and \(T_\varphi\in S^p\). For certain subclasses of \(L^\infty(\mathbb D)\), necessary and sufficient conditions characterizing Schatten class Toeplitz operators are also obtained.
scientific article; zbMATH DE number 5695234
scientific article