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

Abstract

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.