Abstract
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
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.
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.
the Szegö projection \( P_\mathbb{R}^ + \): L2(ℝ)→\( H_ + ^2 \left( \mathbb{R} \right) \) is unitary equivalent to the projection
$$ F:P_\mathbb{R}^ + F^{ - 1} - \chi + I. $$
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© 2008 Birkhäuser Verlag AG
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(2008). Bergman and Poly-Bergman Spaces. In: Commutative Algebras of Toeplitz Operators on the Bergman Space. Operator Theory: Advances and Applications, vol 185. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8726-6_3
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DOI: https://doi.org/10.1007/978-3-7643-8726-6_3
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-8725-9
Online ISBN: 978-3-7643-8726-6
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