Sato Grassmannian中的Hopf-Rinow定理
Hopf-Rinow Theorem in the Sato Grassmannian
摘要：
设$U_2({\cal H})$是希尔伯特空间${\cal H}$中酉算子的Banach-李群，它们是恒等式1的希尔伯特-施密特扰动。本文研究了无限投影$p$在${\cal H}$中的酉轨道$$\{upu^*: u\in U_2({\cal H})\},$$的几何性质。该轨道与对应于极化${\cal H}=p({\cal H})\oplus p({\cal H})^\perp$的希尔伯特-施密特限制格拉斯曼$Gr_{res}(p)$（在文献中也称为佐藤格拉斯曼）中的$p$的连通分量一致。已知$Gr_{res}(p)$的分支是可微流形。在这里，我们给出了一个简单的证明，证明$Gr_{res}^0(p)$是仿射Hilbert空间$p+{\cal B}_2({\cal H})$的光滑子流形，其中${\cal B}_2({\cal H})$表示${\cal H}$的Hilbert-Schmidt算子空间。我们证明了自然联络的测地线，这是$γ(t)=e^{tz}pe^{-tz}$的形式，$z$的$p$ -共对角反厄米元素的${\cal B}_2({\cal H})$，有最小的长度提供$\|z\|\le π/2$。请注意，该条件是根据通常的算子范数给出的，这一事实意味着存在任意长度的极小测地线。我们还证明了任何两点$p_1,p_2\in Gr_{res}^0(p)$连接的最小测地线。如果而且$\|p_1-p_2\|<1$，则极小测地线是唯一的.最后，我们用$k$ -Schatten范数（$k>2$）代替2-范数，并证明了对于这些范数，测地线也是极小的，直到$t$的临界值，这也是根据通常的算子范数估计的。在此过程中，还得到了群$U_2({\cal H})$在$k$ -范数下的极小性结果。
英文摘要：
Let $U_2({\cal H})$ be the Banach-Lie group of unitary operators in the Hilbert space ${\cal H}$ which are Hilbert-Schmidt perturbations of the identity 1. In this paper we study the geometry of the unitary orbit $$\{upu^*: u\in U_2({\cal H})\},$$ of an infinite projection $p$ in ${\cal H}$. This orbit coincides with the connected component of $p$ in the Hilbert-Schmidt restricted Grassmannian $Gr_{res}(p)$ (also known in the literature as the Sato Grassmannian) corresponding to the polarization ${\cal H}=p({\cal H})\oplus p({\cal H})^\perp$. It is known that the components of $Gr_{res}(p)$ are differentiable manifolds. Here we give a simple proof of the fact that $Gr_{res}^0(p)$ is a smooth submanifold of the affine Hilbert space $p+{\cal B}_2({\cal H})$, where ${\cal B}_2({\cal H})$ denotes the space of Hilbert-Schmidt operators of ${\cal H}$. We prove that the geodesics of the natural connection, which are of the form $γ(t)=e^{tz}pe^{-tz}$, for $z$ a $p$-codiagonal anti-hermitic element of ${\cal B}_2({\cal H})$, have minimal length provided that $\|z\|\le π/2$. Note that the condition is given in terms of the usual operator norm, a fact which implies that there exist minimal geodesics of arbitrary length. Also we show that any two points $p_1,p_2\in Gr_{res}^0(p)$ are joined by a minimal geodesic. If moreover $\|p_1-p_2\|<1$, the minimal geodesic is unique. Finally, we replace the 2-norm by the $k$-Schatten norm ($k>2$), and prove that the geodesics are also minimal for these norms, up to a critical value of $t$, which is estimated also in terms of the usual operator norm. In the process, minimality results in the $k$-norms are also obtained for the group $U_2({\cal H})$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
