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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 Open image in new window in terms of a de Branges-type space on Open image in new window . 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.

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Copyright information

© Birkhäuser Verlag AG 2009

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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