$tan 2Θ$定理的推广
A generalization of the $tan 2Θ$ Theorem
摘要：
设$\mathbf{A}$是可分Hilbert空间$\mathfrak{H}$上的有界自伴算子，$\mathfrak{H}_0\subset\mathfrak{H}$是$\mathbf{A}$的闭不变子空间.假设$\sup\spec(A_0)\leq \inf\spec(A_1)$，其中$A_0$和$A_1$分别是$\mathbf{A}$在子空间$\mathfrak{H}_0$和$\mathfrak{H}_1=\mathfrak{H}_0^\perp$上的限制，我们研究了不变子空间$\mathfrak{H}_0$在关于分解$\mathfrak{H} = \mathfrak{H}_0\oplus\mathfrak{H}_1$非对角的有界自伴扰动$\mathbf{V}$下的变化.我们得到了算子$\mathbf{A}$和$\mathbf{B}=\mathbf{A}+\mathbf{V}$的不变子空间上的正交投影的差的范数的尖锐的双边估计。这些结果推广了著名的Davis-Kahan $\tan 2Θ$定理。在此基础上，我们还证明了算子Riccati方程压缩解的新的存在唯一性定理，从而推广了Adamyan，Langer和Tretter最近的结果.
英文摘要：
Let $\mathbf{A}$ be a bounded self-adjoint operator on a separable Hilbert space $\mathfrak{H}$ and $\mathfrak{H}_0\subset\mathfrak{H}$ a closed invariant subspace of $\mathbf{A}$. Assuming that $\sup\spec(A_0)\leq \inf\spec(A_1)$, where $A_0$ and $A_1$ are restrictions of $\mathbf{A}$ onto the subspaces $\mathfrak{H}_0$ and $\mathfrak{H}_1=\mathfrak{H}_0^\perp$, respectively, we study the variation of the invariant subspace $\mathfrak{H}_0$ under bounded self-adjoint perturbations $\mathbf{V}$ that are off-diagonal with respect to the decomposition $\mathfrak{H} = \mathfrak{H}_0\oplus\mathfrak{H}_1$. We obtain sharp two-sided estimates on the norm of the difference of the orthogonal projections onto invariant subspaces of the operators $\mathbf{A}$ and $\mathbf{B}=\mathbf{A}+\mathbf{V}$. These results extend the celebrated Davis-Kahan $\tan 2Θ$ Theorem. On this basis we also prove new existence and uniqueness theorems for contractive solutions to the operator Riccati equation, thus, extending recent results of Adamyan, Langer, and Tretter.
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