特征向量的先验 tanθ 定理
The a priori tanθtheorem for eigenvectors
AI总结：
本文证明当自伴算子A的谱间隙d满足扰动V的范数小于√2 d时，特征向量f与投影到未扰动子空间上的夹角θ满足tanθ ≤ ‖V‖/d，且该界是紧的。
摘要：
设$A$是Hilbert空间$\fH$上的自伴算子。假设$A$的谱由两个不相交的分量$σ_0$和$σ_1$组成，且集合$σ_0$的凸包与$σ_1$不相交。设$V$是$\fH$上的有界自伴算子，关于正交分解$\fH=\fH_0\oplus\fH_1$是非对角元的，其中$\fH_0$和$\fH_1$分别是$A$对应于谱集$σ_0$和$σ_1$的谱子空间。已知如果$\|V\|<\sqrt{2}d$，其中$d=\dist(σ_0,σ_1)>0$，则扰动$V$不会闭合$σ_0$和$σ_1$之间的间隙。假设$f$是扰动算子$A+V$的特征向量，其本征值在区间$(\min(σ_0)-d,\max(σ_0)+d)$内，我们证明在条件$\|V\|<\sqrt{2}d$下，$f$与其在$\fH_0$上的正交投影之间的（锐）角$θ$满足界$\tanθ\leq\frac{\|V\|}{d}$，并且该界是紧的。
英文摘要：
Let $A$ be a self-adjoint operator on a Hilbert space $\fH$. Assume that the spectrum of $A$ consists of two disjoint components $σ_0$ and $σ_1$ such that the convex hull of the set $σ_0$ does not intersect the set $σ_1$. Let $V$ be a bounded self-adjoint operator on $\fH$ off-diagonal with respect to the orthogonal decomposition $\fH=\fH_0\oplus\fH_1$ where $\fH_0$ and $\fH_1$ are the spectral subspaces of $A$ associated with the spectral sets $σ_0$ and $σ_1$, respectively. It is known that if $\|V\|<\sqrt{2}d$ where $d=\dist(σ_0,σ_1)>0$ then the perturbation $V$ does not close the gaps between $σ_0$ and $σ_1$. Assuming that $f$ is an eigenvector of the perturbed operator $A+V$ associated with its eigenvalue in the interval $(\min(σ_0)-d,\max(σ_0)+d)$ we prove that under the condition $\|V\|<\sqrt{2}d$ the (acute) angle $θ$ between $f$ and the orthogonal projection of $f$ onto $\fH_0$ satisfies the bound $\tanθ\leq\frac{\|V\|}{d}$ and this bound is sharp.
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