Nearly invariant and de Branges spaces

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.