Bergman and Poly-Bergman Spaces

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


We start by recalling an old and well-known result. Let \( H_ + ^2 \left( \mathbb{R} \right)\left( { \subset L_2 \left( \mathbb{R} \right)} \right) \) be the Hardy space on the upper half-plane II in ℂ, which by definition consists of all functions ϕ on ℝ admitting analytic continuation in II and satisfying the condition
$$ \mathop {\sup }\limits_{v > 0} \int_{\mathbb{R} + iv} {\left| {\phi \left( {u + iv} \right)} \right|^2 du < \infty .} $$
Let \( P_\mathbb{R}^ + \) be the (orthogonal) Szegö projection of L2(ℝ) onto \( H_ + ^2 \left( \mathbb{R} \right) \). Then: the Fourier transform F gives an isometric isomorphism of the space L 2 (ℝ), under which
  1. 1.
    the Hardy space\( H_ + ^2 \left( \mathbb{R} \right) \) is mapped onto L2(ℝ+),
    $$ F:H_ + ^2 \left( \mathbb{R} \right) \to L_2 \left( {\mathbb{R}_ + } \right), $$
  2. 2.
    the Szegö projection \( P_\mathbb{R}^ + \): L2(ℝ)→\( H_ + ^2 \left( \mathbb{R} \right) \) is unitary equivalent to the projection
    $$ F:P_\mathbb{R}^ + F^{ - 1} - \chi + I. $$


Hardy Space Bergman Space Bergman Kernel Isometric Isomorphism Bergman Projection 
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© Birkhäuser Verlag AG 2008

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