De Branges space

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

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

[edit] 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(Im(z))^{(-1/2)}, \forall z \in \mathbb{C}^+

[edit] 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.

[edit] References

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