arXiv 2008.100582020-08-23math.FA

多区间上希尔伯特变换算子的谱性质

On the spectral properties of the Hilbert transform operator on multi-intervals

Marco Bertola, Alexander Katsevich, Alexander Tovbis

摘要：

设$J,E\subset\mathbb R$是两个内部不相交的多区间。考虑以下运算符$$A:\, L^2( J )\to L^2(E),\ (Af)(x) = \frac 1π\int_{ J } \frac {f(y)\text{d} y}{x-y},$$，并设$A^\dagger$为其伴随。引入了作用于$L^2(E)\oplus L^2(J)$的自伴算子$\mathscr K$，其非对角块由$A$和$A^\dagger$组成.本文研究了$\mathscr K$和算子$A^\dagger A$、$A A^\dagger$的谱性质。我们的主要工具是利用一个适当的Riemann-Hilbert问题得到$\mathscr K$的预解式，记作$\mathscr R$，然后计算$\mathscr R$在谱参数$λ$中的跳跃和极点.证明了$\mathscr K$的谱有绝对连续分量$[0,1]$当且仅当$J$和$E$有公共端点，且其重数等于它们的个数.如果没有公共端点，则$\mathscr K$的谱仅由本征值和$0$组成。如果存在公共端点，则$\mathscr K$可以具有嵌入在连续谱中的本征值，它们中的每一个具有有限的重数，并且本征值可以仅在$0$处累积。在所有情况下，$\mathscr K$不具有奇异连续谱。得到了$A^\dagger A$和$A A^\dagger$的光谱性质，它们与$\mathscr K$的光谱性质非常相似。

英文摘要：

Let $J,E\subset\mathbb R$ be two multi-intervals with non-intersecting interiors. Consider the following operator $$A:\, L^2( J )\to L^2(E),\ (Af)(x) = \frac 1π\int_{ J } \frac {f(y)\text{d} y}{x-y},$$ 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 $λ$. 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.

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