This paper is an effort to continue the legacy of the classically successful theory of Toeplitz operators on the Hardy space over the unit disk to a new domain in \({\mathbb {C}}^d\)—the symmetrized polydisk. We obtain algebraic characterizations of Toeplitz operators, analytic Toeplitz operators, compact perturbation of Toeplitz operators and dual Toeplitz operators. We then revisit the operator theory of this domain considered first in Biswas and Shyam Roy (J Funct Anal 266:6224–6255, 2014), to study the generalized Toeplitz operators and find a commutant lifting type result.

中文翻译：

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对称多圆盘上Toeplitz算子的代数性质

本文致力于将经典成功的Toeplitz算符理论在单位磁盘上的Hardy空间上的遗产延续到\（{\ mathbb {C}} ^ d \）中的新域（对称多磁盘）。我们获得了Toeplitz算子，解析Toeplitz算子，Toeplitz算子和双Toeplitz算子的紧摄动的代数表征。然后，我们重新审视Biswas和Shyam Roy（J Funct Anal 266：6224–6255，2014）中首先考虑的该域的算符理论，以研究广义的Toeplitz算符并找到可交换的提升类型结果。