Motivated by questions related to the compactness of the \({\overline{\partial }}\)-Neumann operator, we study the restriction operator from the Bergman space of a domain in \(\mathbb {C}^n\) to the Bergman space of a non-empty open subset of the domain. We relate the restriction operator to the Toeplitz operator on the Bergman space of the domain whose symbol is the characteristic function of the subset. Using the biholomorphic invariance of the spectrum of the associated Toeplitz operator, we study the restriction operator from the Bergman space of the unit disc to the Bergman space of subdomains with large symmetry groups, such as horodiscs and subdomains bounded by hypercycles. Furthermore, we prove a sharp estimate of the norm of the restriction operator in case the domain and the subdomain are balls. We also study various operator theoretic properties of the restriction operator such as compactness and essential norm estimates.

中文翻译：

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Bergman空间上的约束算子

受与\（{\ overline {\ partial}} \）- Neumann算子的紧致性有关的问题的启发，我们从\（\ mathbb {C} ^ n \）中一个域的Bergman空间研究限制算子域的非空开放子集的Bergman空间。我们将限制算子与域的Bergman空间上的Toeplitz算子相关，该算子的符号是子集的特征函数。利用相关的Toeplitz算子的谱的双全纯不变性，我们研究了从单位圆盘的Bergman空间到具有大对称群的子域（如Horodiscs和以超周期为界的子域）的Bergman空间的限制算子。此外，在域和子域为球形的情况下，我们证明了对限制算子范数的清晰估计。我们还研究了限制算子的各种算子理论特性，例如紧凑性和基本范数估计。