Multiple singular values of Hankel operators

2014/02/07 by Patrick Gérard, Gerard, Patrick, Sandrine Grellier +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Waves and Solitons #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1402.1716

openalex publication_date 2014/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

The goal of this paper is to construct a nonlinear Fourier transformation on the space of symbols of compact Hankel operators on the circle. This transformation allows to solve a general inverse spectral problem involving singular values of a compact Hankel operator, with arbitrary multiplicities. The formulation of this result requires the introduction of the pair made with a Hankel operator and its shifted Hankel operator. As an application, we prove that the space of symbols of compact Hankel operators on the circle admits a singular foliation made of tori of finite or infinite dimensions, on which the flow of the cubic Szegö equation acts. In particular, we infer that arbitrary solutions of the cubic Szegö equation on the circle with finite momentum are almost periodic with values in H^1/2(S ¹).

Citations

• The inverse spectral problem for self-adjoint Hankel operators
• On two problems concerning linear transformations in hilbert space
• On a set of transformations of Gaussian random functions

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