Commuting Toeplitz Operators and Hyperbolic Geometry

Abstract

The C^*-algebras of Toeplitz operators considered in Chapters 5, 6, and 7 are commutative. This property is impossible (except for the trivial case of scalar operators ≡ constant defining symbols) for the C^*-algebras of Toeplitz operators acting on the Hardy space. The special features of the defining symbols which make this phenomenon possible were symbols depending only on the imaginary part of a variable for Toeplitz operators on the upper half-plane, radial symbols for Toeplitz operators on the unit disk, and symbols depending only on the angular part of a variable for Toeplitz operators on the upper half-plane, respectively. In this stage a natural question appears: whether there exist other classes of defining symbols which generate commutative Toeplitz operator C^*-algebras, and how they can be classfied.