A dynamical system approach to the inverse spectral problem for Hankel operators

Liang, Zhehui

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Year:: 2022
Contributor:: Liang, Zhehui (creator); Treil, Sergei (Advisor); Pipher, Jill (Reader); Pausader, Benoit (Reader); Brown University. Department of Mathematics (sponsor)
Genre:: theses
Subject:: Mathematics; Functional analysis
Extent:: xi, 154 p.

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Abstract:: This dissertation is devoted to the study of inverse spectral problem of Hankel operators. It is well-known that spectral characteristics of a Hankel operator does not uniquely define it: there are many unitarily equivalent Hankel operators. However, as it was noticed in breakthrough papers by P.~Gerald and S.~Grellier the spectral characteristics of the Hankel operator and its one column truncation completely determine the compact Hankel operator. This turns out to be true for general Hankel operators, which is one of the results of the thesis. For self-adjoint Hankel operators it is pretty easy to understand what the spectral characteristics are. For the general case the approach is more involved and based on the theory of complex symmetric operators. In both cases we state and prove an abstract theorem that reduces the existence of a Hankel operator with prescribed spectral properties to the asymptotic stability of some contractions, hence the ``dynamical system approach'' in the title. We then apply this abstract theorem to the particular case of compact operators; the asymptotic stability there is obtained almost for free, and the description of spectral characteristics can be greatly simplified. We are able to give new proofs for known results by Gerard--Grillier, as well as to prove some new results. For compact Hankel operator with simple singular values, we will show that it can be uniquely characterized by two sequences of complex numbers whose modulus part satisfy an intertwining relation. While for Hankel operator with non-simple singular values, we will show that it will be uniquely characterized by two sequences of complex numbers with the same requirement, together with two sequences of discrete probability measures on the unit circle.
Notes:: Thesis (Ph. D.)--Brown University, 2022

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Citation

Liang, Zhehui, "A dynamical system approach to the inverse spectral problem for Hankel operators" (2022). Mathematics Theses and Dissertations. Brown Digital Repository. Brown University Library. https://repository.library.brown.edu/studio/item/bdr:tbd3grtj/

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Mathematics Theses and Dissertations

Theses and Dissertations for the Mathematics department.
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