论文标题
Aronszajn-Donoghue定理中有限等级扰动中特殊集的维度
Dimension of the exceptional set in the Aronszajn-Donoghue theorem for finite rank perturbations
论文作者
论文摘要
经典的Aronszajn-Donoghue定理指出，对于自偶会操作员（由循环矢量）的等级，原始操作员的频谱测量的奇异部分是相互奇异的。正如简单的直接总和类型示例所示，此结果不适合有限秩扰动。但是，一组出色的扰动非常小。 也就是说，对于一个等级的家庭，$ d $扰动$ a _ {\boldsymbolα}：= a +\ \ \ \ \ \ \ \ \boldsymbolα\ Mathbf {b}^*$，$ \ mathbf {b} ran $ \，\ mathbf {b} $是$ a $的循环，由$ d \ times d $ d $ hermitian矩阵$ \boldsymbolα$，$ a $ a $ a $ a $ a $和$ a _ _ {\boldsymbolα} $的单个频谱的各个部分是$ $ \ $ \ boldsymbol的$ boldsymbold for a $ bolds smill a a iffience a a _ {\boldsymbolα} $。前两位作者早些时候表明，$ e $是$ d \ times d $ hermitian矩阵的空间$ \ mathbf {h}（d）$的零的子集。 在本文中，我们表明集合$ e $具有小的hausdorff尺寸，$ \ dim e \ e \ dim \ dim \ mathbf {h}（d）（d）-1 = d^2-1 $。
The classical Aronszajn-Donoghue theorem states that for a rank one perturbation of a self-adjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set of exceptional perturbations is pretty small. Namely, for a family of rank $d$ perturbations $A_{\boldsymbolα} := A +\mathbf{B} \boldsymbolα \mathbf{B}^*$, $\mathbf{B}:\mathbb{C}^d\to \mathbf{H}$, with Ran$\,\mathbf{B}$ being cyclic for $A$, parametrized by $d\times d$ Hermitian matrices $\boldsymbolα$, the singular parts of the spectral measures of $A$ and $A_{\boldsymbolα}$ are mutually singular for all $\boldsymbolα$ except for a small exceptional set $E$. It was shown earlier by the first two authors that $E$ is a subset of measure zero of the space $\mathbf{H}(d)$ of $d\times d$ Hermitian matrices. In this paper we show that the set $E$ has small Hausdorff dimension, $\dim E \le \dim\mathbf{H}(d)-1 = d^2-1$.
