arXiv 2008.133102020-08-31math.FA

高阶局部狄利克雷积分与de Branges-Rovnyak空间

Higher order local Dirichlet integrals and de Branges-Rovnyak spaces

Shuaibing Luo, Caixing Gu, Stefan Richter

摘要：

我们研究了扩张Hilbert空间算子$T$，它是等距算子的有限秩扰动。如果$T$的谱包含在闭单位圆盘$\overline{\mathbb{D}}$中，则这种算子的形式为$T= U\oplus R$，其中$U$是等距的，$R$酉等价于de Branges-Rovnyak空间$\mathcal{H}(B)$上的变量$z$的乘法算子。事实上，空间$\mathcal{H}(B)$是根据有理算子值Schur函数$B$定义的。当$\dim \ker T^*=1$，则$\mathcal{H}(B)$可以看作是$\mathbb{D}$中的纯量解析函数空间，并且函数$B$具有由$|B|^2+|a|^2=1$ a.e.关于$\partial \mathbb{D}$。我们证明了一个合理的$B$的配偶$a$是$a(z)=a(0)\frac{p(z)}{q(z)}$的形式，其中$p$和$q$是适当地从两个相关的运营商的特征多项式。若$T$是$2m$ -等距扩张算子，则$p$的所有零点都在单位圆内，并且我们利用在点$w\in \partial \mathbb{D}$上的$m$阶局部Dirichlet积分来完整地描述空间$\mathcal{H}(B)$.

英文摘要：

We investigate expansive Hilbert space operators $T$ that are finite rank perturbations of isometric operators. If the spectrum of $T$ is contained in the closed unit disc $\overline{\mathbb{D}}$, then such operators are of the form $T= U\oplus R$, where $U$ is isometric and $R$ is unitarily equivalent to the operator of multiplication by the variable $z$ on a de Branges-Rovnyak space $\mathcal{H}(B)$. In fact, the space $\mathcal{H}(B)$ is defined in terms of a rational operator-valued Schur function $B$. In the case when $\dim \ker T^*=1$, then $\mathcal{H}(B)$ can be taken to be a space of scalar-valued analytic functions in $\mathbb{D}$, and the function $B$ has a mate $a$ defined by $|B|^2+|a|^2=1$ a.e. on $\partial \mathbb{D}$. We show the mate $a$ of a rational $B$ is of the form $a(z)=a(0)\frac{p(z)}{q(z)}$, where $p$ and $q$ are appropriately derived from the characteristic polynomials of two associated operators. If $T$ is a $2m$-isometric expansive operator, then all zeros of $p$ lie in the unit circle, and we completely describe the spaces $\mathcal{H}(B)$ by use of what we call the local Dirichlet integral of order $m$ at the point $w\in \partial \mathbb{D}$.

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