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 ${\displaystyle \mathbb {C} }$ to ${\displaystyle \mathbb {C} }$ that satisfies the inequality ${\displaystyle |E(z)|>|E({\bar {z}})|}$, for all z in the upper half of the complex plane ${\displaystyle \mathbb {C} ^{+}=\{z\in \mathbb {C} \mid \operatorname {Im} (z)>0\}}$.

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 ^[1]

${\displaystyle B(E)={\bigg \{}F{\text{ is entire }}{\bigg |}{\frac {F}{E}},{\frac {F^{\#}}{E}}\in H_{2}(\mathbb {C} ^{+}),\int _{\mathbb {R} }{\bigg |}{\frac {F(\lambda )}{E(\lambda )}}{\bigg |}^{2}\mathrm {d} \lambda <\infty {\bigg \}}}$

where:

• ${\displaystyle \mathbb {C} ^{+}=\{z\in \mathbb {C} \mid \operatorname {Im} (z)>0\}}$ is the open upper half of the complex plane.
• ${\displaystyle F^{\#}(z)={\overline {F({\bar {z}})}}}$.
• ${\displaystyle H_{2}(\mathbb {C} ^{+})}$ 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:

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

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

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