# Nearly invariant and de Branges spaces

• Alexandru Aleman
• William T. Ross
• Nathan S. Feldman
Part of the Frontiers in Mathematics book series (FM)

## Abstract

It turns out that we can also describe the nearly invariant subspaces of in terms of a de Branges-type space on . First let us review the well-known de Branges spaces on ℂ ∖ ℝ . We follow [25, p. 9–12]. Let Ψ be an analytic function on the upper half plane $$\mathbb{C}_ + = \{ \Im z > 0\}$$ such that ℜΨ≥0. The classical Herglotz theorem [25, p. 7] says that there is a non-negative measure μ on $$\mathbb{R}$$ and a non-negative number p such that
$$\Re \Psi (x + iy) = py + \frac{1} {\pi }\int_{ - \infty }^\infty {\frac{y} {{(t - x)^2 + y^2 }}} d\mu (t), x + iy \in \mathbb{C}_ + .$$
(5.1.1)
The reader will recognize the above integral as the Poisson integral of μ. Extend Ψ to the lower half plane so that
$$\Psi (z) = \overline {\Psi (\bar z),} z = x + iy, y < 0.$$
. A theorem of de Branges [25, p. 9] says that there exists a unique Hilbert space L(Ψ) of analytic functions on $$\mathbb{C}\backslash \mathbb{R}$$ such that for each fixed $$w \in \mathbb{C}\backslash \mathbb{R}$$ , the function
$$z \mapsto \frac{{\Psi (z) + \overline {\Psi (w)} }} {{\pi i(\bar w - z)}}$$
(5.1.2)
belongs to L(Ψ) and
$$F(w) = \left\langle {F(z),\frac{{\Psi (z) + \overline {\Psi (w)} }} {{\pi i(\bar w - z)}}} \right\rangle _{\mathcal{L}(\Psi )} \forall F \in \mathcal{L}(\Psi ).$$
(5.1.3)
The previous identity says that the functions in (5.1.2) are the reproducing kernel functions for L(Ψ). Furthermore, if μ is the measure from (5.1.1), the linear transformation
$$f \mapsto \frac{1} {{\pi i}}\int_{ - \infty }^\infty {\frac{{f(t)}} {{t - z}}} d\mu (t)$$
(5.1.4)
maps L2 (μ) isometrically into L(Ψ) and the orthogonal complement of the range of this transformation contains only constant functions. For example, if p=0 in (5.1.1), this map is onto.

## Authors and Affiliations

• Alexandru Aleman
• 1
• William T. Ross
• 2
• Nathan S. Feldman
• 3
1. 1.Centre for Mathematical SciencesLund UniversityLundSweden
2. 2.Department of Mathematics and Computer ScienceUniversity of RichmondRichmondUSA
3. 3.Department of MathematicsWashington & Lee UniversityLexingtonUSA

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