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 $$\big (H^{\infty }({\mathbb {D}})+C(\overline{{\mathbb {D}}}) \big )\cap \overline{\big (H^{\infty }({\mathbb {D}})+C(\overline{{\mathbb {D}}})\big )}$$ .

中文翻译：

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Bergman-Toeplitz 矩阵的第一 Szegö 定理

在本文中，我们研究了符号在 $$H^{\infty }({\mathbb {D}})+C(\overline{{\mathbb {D}} })$$ 。我们为 Bergman-Toeplitz 算子建立了渐近可逆性的判据和渐近反演公式。这些结果用于获得 Bergman-Toeplitz 矩阵的第一个 Szego 定理的两个版本。此外，我们用符号在 $$\big (H^{\infty }({\mathbb {D}})+C(\overline{{\mathbb {D} }}) \big )\cap \overline{\big (H^{\infty }({\mathbb {D}})+C(\overline{{\mathbb {D}}})\big )}$$ .