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 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 ).

Commutative Toeplitz Algebras and Their Gelfand Theory: Old and New Results

“…3. We present some original results, which extend and generalize for general dimensions the results regarding the case n = 3, which was presented in [24].…”

“…This approach permits also the study of commutative Toeplitz Banach algebras as in the preceding section. We recall the ideas and further extend the analysis in [24].…”

“…It is natural to study the existence and structure of the above commutative Toeplitz algebras for different function Hilbert spaces such as the Hardy space over the unit sphere [24,30] or the Fock space of Gaussian square integrable entire functions [4,13]. In fact, there is an interesting interplay between these cases and the analysis of one of these spaces may be useful in the study of another.…”

“…Section 4 carries out the corresponding analysis in case of the classical Hardy space H 2 (S 2n−1 ) over the unit sphere S 2n−1 in C n , see [1,10,24,30]. An useful ingredient to the proofs is a decomposition of Toeplitz operators (with certain symbols) as an infinite sum of Bergman space Toeplitz operators over B n−1 with integer weights.…”

“…However, there is a different approach based on the known Bergman space theory. In [24], the Hardy space is decomposed into a direct sum of Bergman spaces (with different integer weight parameters). Then results obtained in the Bergman space setting can be applied.…”