谱子空间的先验Tan Theta定理
The a priori Tan Theta Theorem for spectral subspaces
摘要：
设A是可分希尔伯特空间H上的自伴算子。设A的谱由两个不相交的分量s_0和s_1组成，使得集合s_0位于集合s_1的有限间隙中。设V是H上关于分块规范（A）=s_0 \cup s_1的非对角有界自伴算子.已知如果||V|| < \sqrt{2} d，其中d= \dist（s_0，s_1），则扰动V不封闭s_0和s_1之间的间隙，扰动算子L=A+V的谱由分别从s_0和s_1生长的两个孤立分量s '_0和s'_1组成.此外，已知如果V满足更强的界，||V|| < d则下面的sharp norm估计成立：||E_L（s'_0）-E_A（s_0）||\leq sin（arctan（||V||其中E_A（s_0）和E_L（s '_0）分别是与谱集合s_0和s'_0相关联的A和L的谱投影。在目前的工作中，我们证明了这一估计仍然有效和尖锐的d \leq| |V|| < \sqrt{2} d，它完全解决了这个问题。
英文摘要：
Let A be a self-adjoint operator on a separable Hilbert space H. Assume that the spectrum of A consists of two disjoint components s_0 and s_1 such that the set s_0 lies in a finite gap of the set s_1. Let V be a bounded self-adjoint operator on H off-diagonal with respect to the partition spec(A)=s_0 \cup s_1. It is known that if ||V||<\sqrt{2}d, where d=\dist(s_0,s_1), then the perturbation V does not close the gaps between s_0 and s_1 and the spectrum of the perturbed operator L=A+V consists of two isolated components s'_0 and s'_1 grown from s_0 and s_1, respectively. Furthermore, it is known that if V satisfies the stronger bound ||V||< d then the following sharp norm estimate holds: ||E_L(s'_0)-E_A(s_0)|| \leq sin(arctan(||V||/d)), where E_A(s_0) and E_L(s'_0) are the spectral projections of A and L associated with the spectral sets s_0 and s'_0, respectively. In the present work we prove that this estimate remains valid and sharp also for d \leq ||V||< \sqrt{2}d, which completely settles the issue.
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