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European Mathematical Society Publishing House
2024-03-28 20:41:35
2
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=EMSS&vol=3&iss=1&update_since=2024-03-28
EMS Surveys in Mathematical Sciences
EMS Surv. Math. Sci.
EMSS
2308-2151
2308-216X
General
10.4171/EMSS
http://www.ems-ph.org/doi/10.4171/EMSS
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
3
2016
1
CMV matrices, a matrix version of Baxter's theorem, scattering and de Branges spaces
Harry
Dym
The Weizmann Institute of Science, REHOVOT, ISRAEL
David
Kimsey
Ben-Gurion University of the Negev, BEER-SHEVA, ISRAEL
Matrix orthogonal polynomials, Schur parameters CMV matrices, Nehari problem, scattering matrices, reproducing kernel Hilbert spaces, de Branges spaces, Baxter’s theorem
In this survey we establish bijective correspondences between the following classes of objects: (1) $\beta_{-1}$ and $\{ \beta_n \}_{n=0}^{\infty}$, with $\beta_n \in \mathbb C^{p \times p}$ for $n=-1,0,\ldots$, $\beta_{-1}$ unitary, $\| \beta_j \| < 1$ for $j \geq 0$ and $\sum_{j=0}^{\infty} \| \beta_j \| < \infty$; (2) A unitary matrix $\beta_{-1} \in \mathbb C^{p \times p}$ and a spectral density $\Delta$ belonging to the Wiener algebra $\mathcal W^{p \times p}$ with $\Delta(\zeta) \succ 0$ for all $\zeta$ on the unit circle $\mathbb T$; (3) CMV matrices based on a unitary matrix $\beta_{-1} \in \mathbb C^{p \times p}$ and a spectral density $\Delta$ that meets the constraints in (2); (4) scattering matrices that belong to the Wiener algebra $\mathcal W^{p \times p}$; (5) a class of solutions of an associated matricial Nehari problem. The bijective correspondence between summable sequences of contractions and positive spectral densities in the Wiener algebra $\mathcal W^{p \times p}$ (i.e., between class (1) and class (2)) is known as Baxter's theorem and was established by Baxter when $p=1$ and Geronimo when $p \geq 1$. The connections between CMV matrices, the solutions of a related Nehari problem and an inverse scattering problem seem to be new when $p > 1$. There is partial overlap of the connection between the considered Nehari problem and a discrete analogue of an inverse scattering problem considered by Krein and Melik-Adamjan. de Branges spaces of vector-valued polynomials are used to ease a number of computations.
Fourier analysis
Functional analysis
Operator theory
1
105
10.4171/EMSS/14
http://www.ems-ph.org/doi/10.4171/EMSS/14
Topological monsters in elliptic equations and spectral theory
Alberto
Enciso
Consejo Superior de Investigaciones Científicas, MADRID, SPAIN
Daniel
Peralta-Salas
Consejo Superior de Investigaciones Científicas, MADRID, SPAIN
Level sets, elliptic equations, nodal sets, eigenfunctions, harmonic functions
In this paper we will discuss the appearance of complicated geometric and topological structures through the level sets of solutions to a wide range of elliptic equations. Applications to the analysis of nodal sets of eigenfunctions of Schrödinger operators will be discussed too.
Partial differential equations
Global analysis, analysis on manifolds
107
130
10.4171/EMSS/15
http://www.ems-ph.org/doi/10.4171/EMSS/15