Commutative Algebras Generated by Toeplitz Operators on the Unit Sphere

Abstract

The classical result by Brown and Halmos (J Reine Angew Math 213:8–102, 1964) implies that there is no nontrivial commutative $$C^*$$ -algebra generated by Toeplitz operators acting on the Hardy space $$H^2(S^1)$$ , 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 $$C^*$$ 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 $$H^2(S^{2n-1})$$ is drastically different from the one on $$H^2(S^1)$$ . We represent the Hardy space $$H^2(S^{2n-1})$$ as a direct sum of weighted Bergman spaces over $$\mathbb {B}^{n-1}$$ , and use the already known results for the Bergman space operators to describe a variety of nontrivial commutative $$C^*$$ and Banach algebras generated by Toeplitz operators acting on the multidimensional Hardy space $$H^2(S^{2n-1})$$ .