# Bergman and Poly-Bergman Spaces

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

## 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
$$\mathop {\sup }\limits_{v > 0} \int_{\mathbb{R} + iv} {\left| {\phi \left( {u + iv} \right)} \right|^2 du < \infty .}$$
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
$$F:P_\mathbb{R}^ + F^{ - 1} - \chi + I.$$

## Keywords

Hardy Space Bergman Space Bergman Kernel Isometric Isomorphism Bergman Projection
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.