Abstract
A family of recently discovered commutative C ∗ -algebras of Toeplitz operators on the unit disk can be classified as follows. Each pencil of hyperbolic straight lines determines a set of symbols consisting of functions which are constant on the corresponding cycles, the orthogonal trajectories to lines forming a pencil. The C ∗ -algebra generated by Toeplitz operators with such symbols turns out to be commutative. We show that these cases are the only possible ones which generate the commutative C ∗ -algebras of Toeplitz operators on each weighted Bergman space.
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