{"title":"关于Hilbert变换算子在多区间上的谱性质","authors":"M. Bertola, A. Katsevich, A. Tovbis","doi":"10.4171/jst/403","DOIUrl":null,"url":null,"abstract":"Let $J,E\\subset\\mathbb R$ be two multi-intervals with non-intersecting interiors. Consider the following operator $$A:\\, L^2( J )\\to L^2(E),\\ (Af)(x) = \\frac 1\\pi\\int_{ J } \\frac {f(y)\\text{d} y}{x-y},$$ and let $A^\\dagger$ be its adjoint. We introduce a self-adjoint operator $\\mathscr K$ acting on $L^2(E)\\oplus L^2(J)$, whose off-diagonal blocks consist of $A$ and $A^\\dagger$. In this paper we study the spectral properties of $\\mathscr K$ and the operators $A^\\dagger A$ and $A A^\\dagger$. Our main tool is to obtain the resolvent of $\\mathscr K$, which is denoted by $\\mathscr R$, using an appropriate Riemann-Hilbert problem, and then compute the jump and poles of $\\mathscr R$ in the spectral parameter $\\lambda$. We show that the spectrum of $\\mathscr K$ has an absolutely continuous component $[0,1]$ if and only if $J$ and $E$ have common endpoints, and its multiplicity equals to their number. If there are no common endpoints, the spectrum of $\\mathscr K$ consists only of eigenvalues and $0$. If there are common endpoints, then $\\mathscr K$ may have eigenvalues imbedded in the continuous spectrum, each of them has a finite multiplicity, and the eigenvalues may accumulate only at $0$. In all cases, $\\mathscr K$ does not have a singular continuous spectrum. The spectral properties of $A^\\dagger A$ and $A A^\\dagger$, which are very similar to those of $\\mathscr K$, are obtained as well.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2020-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"On the spectral properties of the Hilbert transform operator on multi-intervals\",\"authors\":\"M. Bertola, A. Katsevich, A. Tovbis\",\"doi\":\"10.4171/jst/403\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $J,E\\\\subset\\\\mathbb R$ be two multi-intervals with non-intersecting interiors. Consider the following operator $$A:\\\\, L^2( J )\\\\to L^2(E),\\\\ (Af)(x) = \\\\frac 1\\\\pi\\\\int_{ J } \\\\frac {f(y)\\\\text{d} y}{x-y},$$ and let $A^\\\\dagger$ be its adjoint. We introduce a self-adjoint operator $\\\\mathscr K$ acting on $L^2(E)\\\\oplus L^2(J)$, whose off-diagonal blocks consist of $A$ and $A^\\\\dagger$. In this paper we study the spectral properties of $\\\\mathscr K$ and the operators $A^\\\\dagger A$ and $A A^\\\\dagger$. Our main tool is to obtain the resolvent of $\\\\mathscr K$, which is denoted by $\\\\mathscr R$, using an appropriate Riemann-Hilbert problem, and then compute the jump and poles of $\\\\mathscr R$ in the spectral parameter $\\\\lambda$. We show that the spectrum of $\\\\mathscr K$ has an absolutely continuous component $[0,1]$ if and only if $J$ and $E$ have common endpoints, and its multiplicity equals to their number. If there are no common endpoints, the spectrum of $\\\\mathscr K$ consists only of eigenvalues and $0$. If there are common endpoints, then $\\\\mathscr K$ may have eigenvalues imbedded in the continuous spectrum, each of them has a finite multiplicity, and the eigenvalues may accumulate only at $0$. In all cases, $\\\\mathscr K$ does not have a singular continuous spectrum. The spectral properties of $A^\\\\dagger A$ and $A A^\\\\dagger$, which are very similar to those of $\\\\mathscr K$, are obtained as well.\",\"PeriodicalId\":48789,\"journal\":{\"name\":\"Journal of Spectral Theory\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2020-08-23\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Spectral Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4171/jst/403\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Spectral Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4171/jst/403","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
On the spectral properties of the Hilbert transform operator on multi-intervals
Let $J,E\subset\mathbb R$ be two multi-intervals with non-intersecting interiors. Consider the following operator $$A:\, L^2( J )\to L^2(E),\ (Af)(x) = \frac 1\pi\int_{ J } \frac {f(y)\text{d} y}{x-y},$$ and let $A^\dagger$ be its adjoint. We introduce a self-adjoint operator $\mathscr K$ acting on $L^2(E)\oplus L^2(J)$, whose off-diagonal blocks consist of $A$ and $A^\dagger$. In this paper we study the spectral properties of $\mathscr K$ and the operators $A^\dagger A$ and $A A^\dagger$. Our main tool is to obtain the resolvent of $\mathscr K$, which is denoted by $\mathscr R$, using an appropriate Riemann-Hilbert problem, and then compute the jump and poles of $\mathscr R$ in the spectral parameter $\lambda$. We show that the spectrum of $\mathscr K$ has an absolutely continuous component $[0,1]$ if and only if $J$ and $E$ have common endpoints, and its multiplicity equals to their number. If there are no common endpoints, the spectrum of $\mathscr K$ consists only of eigenvalues and $0$. If there are common endpoints, then $\mathscr K$ may have eigenvalues imbedded in the continuous spectrum, each of them has a finite multiplicity, and the eigenvalues may accumulate only at $0$. In all cases, $\mathscr K$ does not have a singular continuous spectrum. The spectral properties of $A^\dagger A$ and $A A^\dagger$, which are very similar to those of $\mathscr K$, are obtained as well.
期刊介绍:
The Journal of Spectral Theory is devoted to the publication of research articles that focus on spectral theory and its many areas of application. Articles of all lengths including surveys of parts of the subject are very welcome.
The following list includes several aspects of spectral theory and also fields which feature substantial applications of (or to) spectral theory.
Schrödinger operators, scattering theory and resonances;
eigenvalues: perturbation theory, asymptotics and inequalities;
quantum graphs, graph Laplacians;
pseudo-differential operators and semi-classical analysis;
random matrix theory;
the Anderson model and other random media;
non-self-adjoint matrices and operators, including Toeplitz operators;
spectral geometry, including manifolds and automorphic forms;
linear and nonlinear differential operators, especially those arising in geometry and physics;
orthogonal polynomials;
inverse problems.