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
The reader will recognize the above integral as the Poisson integral of μ. Extend Ψ to the lower half plane so that
. 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
belongs to L(Ψ) and
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
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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© 2009 Birkhäuser Verlag AG
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Aleman, A., Ross, W.T., Feldman, N.S. (2009). Nearly invariant and de Branges spaces. In: The Hardy Space of a Slit Domain. Frontiers in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0098-9_5
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DOI: https://doi.org/10.1007/978-3-0346-0098-9_5
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0346-0097-2
Online ISBN: 978-3-0346-0098-9
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