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.

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

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

$${\displaystyle F/E,F^{\#}/E\in H_{2}(\mathbb {C} ^{+})}$$

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.

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

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

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

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.

• Christian Remling (2003). "Inverse spectral theory for one-dimensional Schrödinger operators: the A function". Math. Z. 245: 597–617. doi:10.1007/s00209-003-0559-2.