Rank one perturbations and singular integral operators
Rank one perturbations and singular integral operators
复制标题
一阶扰动和奇异积分算子
DOI：
10.1016/j.jfa.2009.05.008
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发表时间：
2008
影响因子：
1.7
通讯作者：
S. Treil
中科院分区：
文献类型：
--
作者：
C. Liaw;S. Treil
考虑Hilbert空间H上具有循环向量φ∈H−1（A）的自伴算子A的秩一扰动Aα=A+α（φ，φ）φ.扰动算子Aα的谱表示由特殊形式的奇异积分算子给出。这样的运营商表现出我们所谓的“刚性”，并与希尔伯特变换的两个权重估计。证明了Cauchy（Hilbert）变换的两种权估计的一些结果.特别地，证明了正则柯西变换Tε是从L2（μ）到L2（μα）的一致（在ε内）有界算子，其中μ和μα分别是A和Aα的谱测度.作为应用，利用A关于φ的谱测度的密度给出了Aα在闭区间上具有纯绝对连续谱的一个充分条件.考虑了一些例子，例如Jacobi矩阵和具有L2势的薛定谔算子。
We consider rank one perturbations Aα=A+α(⋅,φ)φ of a self-adjoint operator A with cyclic vector φ∈H−1(A) on a Hilbert space H. The spectral representation of the perturbed operator Aαis given by a singular integral operator of special form. Such operators exhibit what we call ‘rigidity’ and are connected with two weight estimates for the Hilbert transform. Also, some results about two weight estimates of Cauchy (Hilbert) transforms are proved. In particular, it is proved that the regularized Cauchy transforms Tεare uniformly (in ε) bounded operators from L2(μ) to L2(μα), where μ and μαare the spectral measures of A and Aα, respectively. As an application, a sufficient condition for Aαto have a pure absolutely continuous spectrum on a closed interval is given in terms of the density of the spectral measure of A with respect to φ. Some examples, like Jacobi matrices and Schrödinger operators with L2potentials are considered.