Toeplitz Operators on the Upper Half-Plane with Homogeneous Symbols

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


In this chapter we return to the upper half-plane Π, the space L2(Π) and its Bergman subspace A2(Π). Passing to polar coordinates we have
$$ L_2 \left( \prod \right) = L_2 \left( {\mathbb{R}_ + ,rdr} \right) \otimes L_2 \left( {\left[ {0,\pi } \right],d\theta } \right) = L_2 \left( {\mathbb{R}_ + ,rdr} \right) \otimes L_2 \left( {0,\pi } \right), $$
$$ \frac{\partial } {{\partial \bar z}} = \frac{{\cos \theta + i\sin \theta }} {2}\left( {\frac{\partial } {{\partial r}} + i\frac{1} {r}\frac{\partial } {{\partial \theta }}} \right) = \frac{{\cos \theta + i\sin \theta }} {{2r}}\left( {r\frac{\partial } {{\partial r}} + i\frac{\partial } {{\partial \theta }}} \right). $$


Boundary Point Compact Operator Toeplitz Operator Local Algebr Bergman Projection 
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© Birkhäuser Verlag AG 2008

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