# Bergman and Poly-Bergman Spaces

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