An inverse problem for a class of canonical systems having Hamiltonians of determinant one
Section snippets
Introduction and results
The primary motivation for the present paper is to prepare the ground for the subsequent paper [28], where a new equivalent condition for the Grand Riemann Hypothesis for a wide class of zeta or L-functions in number theory is established in terms of an inverse problem for canonical systems. However, the relationship between zeta or L-functions and an inverse problem addressed there is not considered a matter specific to number theory and can be generalized due to its intrinsic interest. With
-spaces and Fourier transform
Let be the -space on an interval I with respect to the Lebesgue measure. If , we regard as a subspace of by the extension by zero. We denote by the Fourier integral and inverse Fourier integral, respectively. We use the same notation for the Fourier transforms on and if no confusion arises. If we understand the right-hand sides in -sense, they provide isometries on up to a constant multiple:
Theory of de Branges spaces
In this section, we review several basic notions and properties of de Branges spaces as a preparation for the next section. A general theory of de Branges spaces is given in the book [7], but the proof of results is more accessible in de Branges' earlier papers [2], [3], [4], [5], [6]. See also [32], [33] and references therein. For the proof of Theorem dB, [23] is also helpful.
Proof of Theorem 1.2
Throughout this section, we suppose that E belongs to , satisfies (K1)∼(K5), and in (K5). Then is inner in , defines an isometry on satisfying by Lemma 2.1, Lemma 2.2. Under the above setting, we study de Branges subspaces of to prove Theorem 1.2.
Inner property and isometry
We prove that the converse of Lemma 2.2 holds in the following sense. For , the multiplication defines a map from into by (1.5). We denote it by and define where . If Θ is an inner function in , images and are subspaces of and , respectively. Obviously, the map is related to the function K by (K2).
Lemma 5.1 Let θ be an analytic function defined in . Suppose that for every . Then the pointwise
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