arXiv 0810.27502008-10-15math.FAmath.CA

秩一扰动与奇异积分算子

Rank one perturbations and singular integral operators

Constanze Liaw, Sergei Treil

备注：: 24 pages
发表信息：: J. Funct. Anal. 257 (2009), no. 6, 1947--1975
DOI：: 10.1016/j.jfa.2009.05.008

摘要：

考虑Hilbert空间$\mathcal H$上具有循环向量$φ\in\mathcal H_{-1}(A)$的自伴算子$A$的秩一扰动$A_α=A+α(\cdot,φ)φ$.扰动算子$A_α$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$T_\varepsilon$是从$L^2(μ)$到$L^2(μ_α)$的一致（在$\varepsilon$中）有界算子，其中$μ$和$μ_α$分别是$A$和$A_α$的谱测度.作为应用，给出了$A_α$在闭区间上具有纯绝对连续谱的一个充分条件.一些例子，如雅可比矩阵和薛定谔算子与$L^2$潜力被认为是。

英文摘要：

We consider rank one perturbations $A_α=A+α(\cdot,φ)φ$ of a self-adjoint operator $A$ with cyclic vector $φ\in\mathcal H_{-1}(A)$ on a Hilbert space $\mathcal H$. The spectral representation of the perturbed operator $A_α$ is 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 $T_\varepsilon$ are uniformly (in $\varepsilon$) bounded operators from $L^2(μ)$ to $L^2(μ_α)$, where $μ$ and $μ_α$ are the spectral measures of $A$ and $A_α$, respectively. As an application, a sufficient condition for $A_α$ to 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 $L^2$ potentials are considered.

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摘要：

英文摘要：

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