# De Branges space

In mathematics, a de Branges space (sometimes written De Branges space) is a concept in functional analysis and is constructed from a de Branges function.

The concept is named after Louis de Branges who proved numerous results regarding these spaces, especially as Hilbert spaces, and used those results to prove the Bieberbach conjecture.

## De Branges functions

A de Branges function is an entire function E from $\mathbb{C}$ to $\mathbb{C}$ that satisfies the inequality $|E(z)| > |E(\bar z)|$, for all z in the upper half of the complex plane $\mathbb{C}^+ = \{z \in \mathbb{C} | {\rm Im}(z) > 0\}$.

## Definition 1

Given a de Branges function E, the de Branges space B(E) is defined as the set of all entire functions F such that

$F/E,F^{\#}/E \in H_2(\mathbb{C}^+)$

where:

• $\mathbb{C}^+ = \{z \in \mathbb{C} | {\rm Im(z)} > 0\}$ is the open upper half of the complex plane.
• $F^{\#}(z) = \overline{F(\bar z)}$.
• $H_2(\mathbb{C}^+)$ is the usual Hardy space on the open upper half plane.

## Definition 2

A de Branges space can also be defined as all entire functions F satisfying all of the following conditions:

• $\int_{\mathbb{R}} |(F/E)(\lambda)|^2 d\lambda < \infty$
• $|(F/E)(z)|,|(F^{\#}/E)(z)| \leq C_F(\operatorname{Im}(z))^{(-1/2)}, \forall z \in \mathbb{C}^+$

## As Hilbert spaces

Given a de Branges space B(E). Define the scalar product:

$[F,G]=\frac{1}{\pi} \int_{\mathbb{R}} \overline{F(\lambda)} G(\lambda) \frac{d\lambda}{|E(\lambda)|^2}.$

A de Branges space with such a scalar product can be proven to be a Hilbert space.

## References

• Christian Remling (2003). "Inverse spectral theory for one-dimensional Schrödinger operators: the A function". Math. Z. 245.