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

Part of the Operator Theory: Advances and Applications book series (OT, volume 185)


Recall that by Theorems 10.4.9 and 10.5.1, the function
$$ \begin{gathered} \gamma a,\lambda \left( t \right) = \frac{{t^{\lambda + 1} }} {{\Gamma \left( {\lambda + 1} \right)}}\int_0^\infty {a(\frac{\eta } {2})\eta ^\lambda e^{ - t\eta } d\eta } \hfill \\ = \frac{1} {{\Gamma \left( {\lambda + 1} \right)}}\int_0^\infty {a(\frac{\eta } {{2t}})\eta ^\lambda e^{ - \eta } d\eta } \hfill \\ \end{gathered} $$
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, ∞).


Toeplitz Operator Straight Line Segment Slow Oscillation Parabolic Case Bounded Symbol 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Birkhäuser Verlag AG 2008

Personalised recommendations