Dynamics of Properties of Toeplitz Operators on the Upper Half-Plane: Hyperbolic Case

Abstract

Recall that by Theorems 10.4.16 and 10.5.1, the function $$ \gamma a,\lambda \left( \xi \right) = \left( {\int_0^\pi {e^{ - 2\xi \theta } \sin ^\lambda \theta d\theta } } \right)^{ - 1} \int_0^\pi {a\left( \theta \right)e^{ - 2\xi \theta } \sin ^\lambda \theta d\theta } , \xi \in \mathbb{R} $$ (14.1.1) is responsible for the boundedness of a Toeplitz operator with symbol a(θ) (∈L1(0, π)). If a(θ)∈L∞(0, π), then the operator T a (λ) is obviously bounded on all spaces A λ 2 (Π), for λ∈(−1, ∞), and the corresponding norms are uniformly bounded by sup z |a(z)|. That is, all spaces A λ 2 (Π), where λ∈(−1, ∞), are natural and appropriate for Toeplitz operators with bounded symbols. Studying unbounded symbols, we wish to have a sufficiently large class of them common to all admissible λ; moreover, we are especially interested in properties of Toeplitz operators for large values of λ. Thus it is convenient for us to consider λ belonging only to [0, ∞), which we will always assume in what follows.