The classical result by Brown and Halmos (J Reine Angew Math 213:8–102, 1964) implies that there is no nontrivial commutative $$C^*$$ C ∗ -algebra generated by Toeplitz operators acting on the Hardy space $$H^2(S^1)$$ H 2 ( S 1 ) , while there are only two commutative Banach algebras. One of them is generated by Toeplitz operators with analytic symbols, and the other one is generated by Toeplitz operators with conjugate analytic symbols. At the same time there are many nontrivial commutative $$C^*$$ C ∗ and Banach algebras generated by Toeplitz operators acting on the Bergman spaces. In the paper we show that the situation on the multidimensional Hardy space $$H^2(S^{2n-1})$$ H 2 ( S 2 n - 1 ) is drastically different from the one on $$H^2(S^1)$$ H 2 ( S 1 ) . We represent the Hardy space $$H^2(S^{2n-1})$$ H 2 ( S 2 n - 1 ) as a direct sum of weighted Bergman spaces over $$\mathbb {B}^{n-1}$$ B n - 1 , and use the already known results for the Bergman space operators to describe a variety of nontrivial commutative $$C^*$$ C ∗ and Banach algebras generated by Toeplitz operators acting on the multidimensional Hardy space $$H^2(S^{2n-1})$$ H 2 ( S 2 n - 1 ) .

中文翻译：

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单位球面上托普利兹算子生成的交换代数

Brown 和 Halmos 的经典结果 (J Reine Angew Math 213:8-102, 1964) 意味着不存在非平凡的交​​换 $$C^*$$ C ∗ - 代数由 Toeplitz 算子作用于 Hardy 空间 $$H ^2(S^1)$$ H 2 ( S 1 ) ，而只有两个交换巴拿赫代数。其中一个是由带有解析符号的托普利兹算子生成的，另一个是由带有共轭解析符号的托普利兹算子生成的。同时，还有许多非平凡的可交换 $$C^*$$C ∗ 和 Banach 代数由作用于 Bergman 空间的 Toeplitz 算子生成。在论文中，我们表明多维哈代空间 $$H^2(S^{2n-1})$$ H 2 ( S 2 n - 1 ) 上的情况与 $$H^2 上的情况截然不同(S^1)$$ H 2 ( S 1 ) 。