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

Abstract

Recall that by Theorems 10.4.9 and 10.5.1, the function $$ \begin{gathered} \gamma a,\lambda \left( t \right) = \frac{{t^{\lambda + 1} }} {{\Gamma \left( {\lambda + 1} \right)}}\int_0^\infty {a(\frac{\eta } {2})\eta ^\lambda e^{ - t\eta } d\eta } \hfill \\ = \frac{1} {{\Gamma \left( {\lambda + 1} \right)}}\int_0^\infty {a(\frac{\eta } {{2t}})\eta ^\lambda e^{ - \eta } d\eta } \hfill \\ \end{gathered} $$ (13.1.1) is responsible for the boundedness of a Toeplitz operator with symbol a=a(y). If \( a = a\left( y \right) \in L_\infty \left( {\mathbb{R}_ + } \right) \), then the operator Ta(λ) is obviously bounded on all spaces \( \mathcal{A}_\lambda ^2 \left( \Pi \right) \), where λ∈(−1, ∞), and the corresponding norms are uniformly bounded by sup z |a(z)|. That is, all spaces \( \mathcal{A}_\lambda ^2 \left( \Pi \right) \), where λ∈(−1, ∞), are natural and appropriate for Toeplitz operators with bounded symbols. One of our aims is a systematic study of unbounded symbols. To avoid unnecessary technicalities in this chapter we will always assume that λ∈[0, ∞).