Parabolic Quasi-radial Quasi-homogeneous Symbols and Commutative Algebras of Toeplitz Operators

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We describe new Banach (not C* !) algebras generated by Toeplitz operators which are commutative on each weighted Bergman space over the unit ball \( \mathbb{B}^n \) , where n > 2. For n = 2 all these algebras collapse to the single C*-algebra generated by Toeplitz operators with quasi-parabolic symbols. As a by-product, we describe the situations when the product of mutually commuting Toeplitz operators is a Toeplitz operator itself.

This work was partially supported by CONACYT Project 80503, México.
Communicated by I.M. Spitkovsky.