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L. Branges (1965)
Some Hilbert spaces of analytic functions. IIIJournal of Mathematical Analysis and Applications, 11
D. Arov, H. Dym (2005)
Strongly Regular J-inner Matrix-valued Functions and Inverse Problems for Canonical Systems
H. Dym, H. McKean (1976)
Gaussian processes, function theory, and the inverse spectral problem
L. Branges (1968)
Hilbert spaces of entire functions
L. Branges (1965)
SOME HILBERT SPACES OF ANALYTIC FUNCTIONS. PART II
L. Branges (1963)
Some Hilbert spaces of analytic functions. ITransactions of the American Mathematical Society, 106
H. Langer, M. Langer, Z. Sasvári (2004)
Continuations of Hermitian indefinite functions and corresponding canonical systems: an exampleMethods of Functional Analysis and Topology, 10
E. Titchmarsh (1938)
Introduction to the Theory of Fourier Integrals
D. Alpay, H. Dym (1985)
Hilbert spaces of analytic functions, inverse scattering and operator models.IIIntegral Equations and Operator Theory, 8
M. Brodskiĭ (1971)
Triangular and Jordan representations of linear operators
D. Arov, H. Dym (1997)
J-inner matrix functions, interpolation and inverse problems for canonical systems, I: FoundationsIntegral Equations and Operator Theory, 29
D. Arov, H. Dym (2012)
Bitangential Direct and Inverse Problems for Systems of Integral and Differential Equations
H. Dym (1970)
An introduction to de Branges spaces of entire functions with applications to differential equations of the Sturm-Liouville typeAdvances in Mathematics, 5
L. Sakhnovich (1997)
Interpolation Theory and Its Applications
D. Arov, H. Dym (2008)
J-Contractive Matrix Valued Functions and Related Topics
[In the study of the class U((j)) of mvf’s (matrix-valued functions) that are ??-inner with respect to the open upper half-plane C+ and a given signature matrix j, special roles are played by the classes uer((j)), (j)(Ur), UrR((j)) and () of left regular, right regular, left strongly regular and right strongly regular j-inner mvf’s. These are discussed at length in [ArD08] and the references cited therein. Shorter introductions may be found in the survey articles [ArD05] and [ArD07]. In particular, these classes are characterized in terms of the RKHS’s (reproducing kernel Hilbert spaces) H(u) that are associated with each mvf u ∈u(j)(??). If u = u1u2 with u1, u2 u(j)(), then, by a theorem of L. de Branges, the RKHS H(H1) is contractively included in H(u); necessary and sufficient conditions for isometric inclusion are also given. In this paper we introduce the class uBr((j)) of B-regular j-inner mvf’s. It is characterized by the fact that if u = u1u2 with u1, u2 u(j), then H(u1) is isometrically included in H(i). If u((j)) is the characteristic mvf of a Livsic-Brodskii operator node, i.e., if u is holomorphic at the point ?? = 0 and normalized by ??(0) = ????, then, thanks to another theorem of L. de Branges, u(uBr) if and only if every normalized left divisor u1 of u(u(j)) is left regular in the Brodskii sense. We shall show that ul(u) ues(r(j)) ubr((j)). We shall also discuss the inverse monodromy problem for canonical differential systems for monodromy matrices U(ubr) and shall present an example of a 2×2 canonical differential system for which the matrizant (fundamental solution) u belongs to the class ul(j) for every × > 0, but does not belong to the class urb((j)).]
Published: Jan 3, 2012
Keywords: Canonical systems, de Branges spaces; J-inner matrix-valued functions; reproducing kernel Hilbert spaces; Livsic-Brodskii nodes.
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