De Branges space
# 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 Hermite-Biehler function, also known as 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 $\mathbb {C} ^{+}=\{z\in \mathbb {C} \mid \operatorname {Im} (z)>0\}$.

## Definition 1

Given a Hermite-Biehler 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:

* $\mathbb {C} ^{+}=\{z\in \mathbb {C} \mid \operatorname {Im} (z)>0\}$ 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:

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

## Definition 3

There exists also an axiomatic description, useful in operator theory.

## 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.

