Commutative Algebras of Toeplitz Operators

Abstract

The commutative C^*-algebras of Toeplitz operators on the classical (weightless) Bergman space were classified in Chapter 9 by pencils of geodesics on the unit disk, considered as the hyperbolic plane. Theorem 10.4.1 shows that the same classes of defining symbols generate commutative C^*-algebras of Toeplitz operators on each weighted Bergman space. At the same time the principal question, whether the above cases are the only possible sets of defining symbols which might generate the commutative C^*-algebras of Toeplitz operators on each weighted Bergman space, has remained open.