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Commutative \(C^*\)-algebras of Toeplitz operators on the unit ball. I: Bargmann-type transforms and spectral representations of Toeplitz operators. (English) Zbl 1144.47024

The main result of [S.Grudsky, R.Quiroga–Barranco, and N.Vasilevski, J. Funct.Anal.234, No.1, 1–44 (2006; Zbl 1100.47023)] says that the \(C^*\)-algebra generated by Toeplitz operators is commutative on each commonly considered Bergman space over the unit disk in \({\mathbb C}\) if and only if there is a maximal commutative subgroup of the Möbius transformation such that the symbols of the Toeplitz operators are invariant under the action of this subgroup. The paper under review and its sequel [Integral Equations Oper.Theory 60, No.1, 89–132 (2008; Zbl 1144.47025), see the following review] aim at extending the above mentioned results from the unit disk in \({\mathbb C}\) to the unit ball \({\mathbb B}^n\) in \({\mathbb C}^n\).
In the present paper, it is shown that there exist \(n+2\) different cases of commutative \(C^*\)-algebras generated by Toeplitz operators on weighted Bergman spaces over \({\mathbb B}^n\). For \(n=1\), these algebras coincide exactly with the three known types of the commutative algebras on the unit disk. In all cases, the bounded measurable symbols of Toeplitz operators are invariant under the action of certain commutative subgroups of biholomorphisms of \({\mathbb B}^n\).

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47L80 Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.)
32A36 Bergman spaces of functions in several complex variables
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