De Branges space

De Branges functions

A Hermite-Biehler function, also known as de Branges function is an entire function E from \Complex to \Complex that satisfies the inequality |E(z)| |E(\bar z)|, for all z in the upper half of the complex plane \Complex^+ = {z \in \Complex \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

B(E) = \bigg { F \text{ is entire } \bigg | \frac{F}{E}, \frac{F^{#}}{E} \in H_2(\Complex^+), \int_{\reals} \bigg | \frac{F(\lambda)}{E(\lambda)} \bigg |^2 \mathrm{d}\lambda

where:

• \Complex^+ = {z \in \Complex \mid \operatorname{Im}(z) 0} is the open upper half of the complex plane.
• F^{#}(z) = \overline{F(\bar z)}.
• H_2(\Complex^+) 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_{\Reals} |(F/E)(\lambda)|^2 d\lambda
• |(F/E)(z)|,|(F^{#}/E)(z)| \leq C_F(\operatorname{Im}(z))^{(-1/2)}, \forall z \in \Complex^+

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_{\Reals} \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

References

1. Mikhaylov, Alexander. (2018-04-15). "The boundary control method and de Branges spaces. Schrödinger equation, Dirac system and discrete Schrödinger operator". Journal of Mathematical Analysis and Applications.