Spectral properties of some unions of linear spaces

区间并集与局部平移群的谱性质

Spectral properties of unions of intervals and groups of local translations

研究有限区间并集上微分算子的自伴扩张对应的单参数酉群，给出其公式并用于刻画允许指数函数正交基的集合的几何性质。

Unitary groups and spectral sets

We study spectral theory for bounded Borel subsets of $\br$ and in particular finite unions of intervals. For Hilbert space, we take $L^2$ of the union of the intervals. This yields a boundary value problem arising from the minimal operator $\Ds = \frac1{2πi}\frac{d}{dx}$ with domain consisting of $C^\infty$ functions vanishing at the endpoints. We offer a detailed interplay between geometric configurations of unions of intervals and a spectral theory for the corresponding selfadjoint extensions of $\Ds$ and for the associated unitary groups of local translations. While motivated by scattering…

乘积谱测度因子的谱性

Spectrality of factors of product spectral measures

改进了Greenfeld和Lev的乘积谱集方法，将定理推广到奇异测度，并建立了新的区间谱并集类，使乘积谱集问题有肯定答案。

Frame spectral pairs and exponential bases

Given a domain $Ω\subset\Bbb R^d$ with positive and finite Lebesgue measure and a discrete set $Λ\subset \Bbb R^d$, we say that $(Ω, Λ)$ is a {\it frame spectral pair} if the set of exponential functions $\mathcal E(Λ):=\{e^{2πi λ\cdot x}: λ\in Λ\}$ is a frame for $L^2(Ω)$. Special cases of frames include Riesz bases and orthogonal bases. In the finite setting $\Bbb Z_N^d$, $d, N\geq 1$, a frame spectral pair can be similarly defined. %(Here, $\Bbb Z_N$ is the cyclic abelian group of order.) We show how to construct and obtain new classes of frame spectral pairs in $\Bbb R^d$ by "adding" frame…

调和Bergman空间上具有多项式符号的Toeplitz算子的谱结构

The spectral structure of Toeplitz operators with polynomial symbols on the harmonic Bergman space

本文利用对称多项式理论，证明了调和Bergman空间上解析多项式符号的Toeplitz算子谱为闭曲线与有限个孤立特征值之并，从而验证了Weyl定理。

酉群的谱隙性质：围绕 Rider 关于非交换 Sidon 集的结果

Spectral gap properties of the unitary groups: around Rider's results on non-commutative Sidon sets

本文证明了 Rider 关于非交换紧群对偶中 Sidon 集合并集与随机 Sidon 集的未发表结果，关键在于利用 Weyl 特征标公式证明酉群的一致谱隙性质，并给出了适用于矩阵值函数正交系的新证明与定量估计。