论文标题
高级当地的迪利奇集体和de branges-rovnyak空间
Higher order local Dirichlet integrals and de Branges-Rovnyak spaces
论文作者
论文摘要
我们研究了宽敞的希尔伯特太空运营商$ t $，这些$ t $是等距运算符的有限排名扰动。如果$ t $的频谱包含在封闭的单元圆盘$ \ etproline {\ mathbb {d}} $中，那么此类运营商的形式为$ t = u \ oplus r $，其中$ u $是等距的，$ r $是单位等于乘以可变$ z $ z $的branges $ $ $ $ $ y $ $ z $ $ brange $ $ $ brange $ $ $ brange-bovny $ f。实际上，空间$ \ MATHCAL {H}（B）$是根据有理运算符值Schur函数$ B $定义的。如果$ \ dim \ ker t^*= 1 $，则可以将$ \ mathcal {h}（b）$视为$ \ mathbb {d} $中标量的分析函数的空间，并且功能$ b $具有由$ | b | b | b |^2+| a | a | a | a | a | a | a | a |^2 = 1 $ a.e定义的函数$ b $。在$ \ partial \ mathbb {d} $上。我们向$ a（z）= a（z）\ frac {p（z）} {q（z）} $的伴侣$ a $ a $ a $ a $ a $ a $ a $ a（z）= a（z）= a（z）\ frac {p（z）} $，其中$ p $ and $ p $和$ q $是从两个相关操作员的特征多项式衍生得出的。如果$ t $是$ 200M $ - 等级扩张的运算符，则$ p $的所有零位于单位圆圈中，我们完全描述了空间$ \ nathcal {h}（b）$，通过使用我们所谓的local dirichlet $ m $ n of Point $ w \ in \ in \ in \ int \ pottial \ Mathbb {d d} $。
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 $w\in \partial \mathbb{D}$.
