The First Szegö Theorem for the Bergman–Toeplitz Matrix

Abstract

In this paper, we study the asymptotic behavior of the determinants of Bergman–Toeplitz matrices with symbols in $$H^{\infty }({\mathbb {D}})+C(\overline{{\mathbb {D}}})$$ . We establish a criterion of the asymptotic invertibility and an asymptotic inversion formula for Bergman–Toeplitz operators. These results are applied to obtain two versions of the first Szego theorem for Bergman–Toeplitz matrices. Moreover, we describe the asymptotic distribution of singular values of Bergman–Toeplitz matrices with symbols in $$\big (H^{\infty }({\mathbb {D}})+C(\overline{{\mathbb {D}}}) \big )\cap \overline{\big (H^{\infty }({\mathbb {D}})+C(\overline{{\mathbb {D}}})\big )}$$ .