A geometric approach to finite rank unitary perturbations
A geometric approach to finite rank unitary perturbations
复制标题
有限阶酉扰动的几何方法
DOI：
10.1512/iumj.2013.62.5028
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发表时间：
2011
影响因子：
1.1
通讯作者：
C. Liaw
中科院分区：
文献类型：
--
作者：
R. Douglas;C. Liaw
罗纳德湾道格拉斯和康斯坦茨廖摘要。对于固定的n∈ N，考虑可分Hilbert空间上亏指数为（n，n）的完全非酉压缩（cnu）的秩n酉扰动族，将这种压缩的酉膨胀与它的秩n酉扰动联系起来.在此基础上，我们证明了扰动算子的谱是纯奇异的当且仅当对应于未扰动算子的算子值特征函数是内函数.在n= 1的情况下，后一种说法归结为秩一微扰理论中的一个著名结果。然而，我们的证明方法，通过理论的伸缩扩展到任意的情况下，n∈ N。我们找到了一个公式的算子值的特征函数对应于一个家庭的相关cnu压缩。当n= 1时，原压缩的特征函数，我们得到了一个关于某个测度的规范化柯西变换的简单表达式。这种表示的应用，然后使我们能够控制跳跃行为的规范柯西变换“跨越”单位圆。
RONALD G. DOUGLAS AND CONSTANZE LIAWAbstract. For fixed n∈ N, we consider a family of rank nunitary perturbations of acompletely non-unitary contraction (cnu) with deficiency indices (n,n) on a separableHilbert space.We relate the unitary dilation of such a contraction to its rank n unitary per-turbations. Based on this construction, we prove that the spectra of the perturbedoperators are purely singular if and only if the operator-valued characteristic func-tion corresponding to the unperturbed operator is inner. In the case where n= 1 thelatter statement reduces to a well-known result in the theory of rank one perturba-tions. However, our method of proof via the theory of dilations extends to the caseof arbitrary n∈ N.We find a formula for the operator-valued characteristic functions correspondingto a family of related cnu contractions. In the case where n= 1, the characteristicfunction of the original contraction we obtain a simple expression involving the nor-malized Cauchy transform of a certain measure. An application of this representationthen enables us to control the jump behavior of this normalized Cauchy transform“across” the unit circle.