The distinguished boundary $b\Gamma$ of the symmetrized bidisc is topologically identifiable with the Mobius strip and it is natural to consider bounded measurable functions there. In this article, we show that there is a natural Hilbert space $H^2(\mathbb G)$. We describe three isomorphic copies of this space. The $L^\infty$ functions on $b\Gamma$ induce Toeplitz operators on this space. Such Toeplitz operators can be characterized through a couple of relations that they have to satisfy with respect to the co-ordinate multiplications on the space $H^2(\mathbb G)$ which we call the Brown-Halmos relations. A number of results are obtained about the Toeplitz operators which bring out the similarities and the differences with the theory of Toeplitz operators on the disc as well as the bidisc. We show that the Coburn alternative fails, for example. However, the compact perturbations of Toeplitz operators are precisely the asymptotic Toeplitz operators. This requires us to find a characterization of compact operators on the Hardy space $H^2(\mathbb G)$. The only compact Toeplitz operator turns out to be the zero operator. We study dual Toeplitz operators in the last section of this paper. In that section, we produce a new result about a family of commuting $\Gamma$-isometries. Just like a Toeplitz operator is characterized by the Brown-Halmos relations with respect to the co-ordinate multiplications, an arbitrary bounded operator $X$ which satisfies the Brown-Halmos relations with respect to a commuting family of $\Gamma$-isometries is a compression of a norm preserving $Y$ acting on the space of minimal $\Gamma$-unitary extension of the family of isometries. Moreover, if $X$ commutes with the $\Gamma$-isometries, then $Y$ is an extension and commutes with the minimal $\Gamma$-unitary extensions. This result is then applied to characterize a dual Toeplitz operator.

中文翻译：

---

对称 Bidisc 上的 Toeplitz 算子

对称双圆盘的可分辨边界 $b\Gamma$ 在拓扑上可与莫比乌斯带相识别，并且在那里考虑有界可测函数是很自然的。在本文中，我们证明存在一个自然的希尔伯特空间 $H^2(\mathbb G)$。我们描述了这个空间的三个同构副本。$b\Gamma$ 上的 $L^\infty$ 函数在这个空间上引入 Toeplitz 算子。这些托普利兹算子可以通过几个关系来表征，它们必须满足空间 $H^2(\mathbb G)$ 上的坐标乘法，我们称之为 Brown-Halmos 关系。得到了一些关于托普利兹算子的结果，指出了托普利兹算子与圆盘和双盘上的托普利兹算子理论的异同。例如，我们表明 Coburn 替代方案失败了。然而，Toeplitz 算子的紧致扰动正是渐近 Toeplitz 算子。这要求我们在 Hardy 空间 $H^2(\mathbb G)$ 上找到紧算子的表征。唯一的紧凑 Toeplitz 算子是零算子。我们在本文的最后一节研究了对偶 Toeplitz 算子。在该部分中，我们产生了关于通勤 $\Gamma$-isometries 族的新结果。就像 Toeplitz 算子的特征在于关于坐标乘法的 Brown-Halmos 关系，满足关于 $\Gamma$-isometries 的通勤族的 Brown-Halmos 关系的任意有界算子 $X$ 是作用于最小 $\Gamma$ 空间的范数保留 $Y$ 的压缩 - 等轴测族的幺正扩展。而且，如果 $X$ 与 $\Gamma$-isometries 交换，则 $Y$ 是一个扩展并且与最小的 $\Gamma$-unitary 扩展交换。然后将此结果应用于表征双 Toeplitz 算子。