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 Bn, 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.