论文标题
基于驱动歧管上的分析捆绑结构
Analytic bundle structure on the idempotent manifold
论文作者
论文摘要
令$ x $为（真实或复杂的）Banach空间，而$ \ Mathcal {i}（x）$是所有（非零和非身份）IDEMPOTENTS的集合；即，在$ x $上的有限线性运算符，其正方形等于自己。我们表明，$ \ Mathcal {i}（x）$的Banach Submanifold $ \ Mathcal {l}（x）$是一个本地微不足道的分析仿射式捆绑包，这是通过发送$ q $ q in $ q in $ q in $ q in的Grassmann歧管$ \ MATHSCR {g}（x）$每只光纤上的仿生式空间结构是从$ \ Mathcal {l}（x）$引起的（尤其是，每个光纤都是$ \ Mathcal {l}（x）$）的仿射式子空间。 使用此信息，我们表明，如果$ k $是真正的希尔伯特空间，则分配$$（e，t）\ mapsto t^*\ circ p_ {e^\ bot} + p_ {e}，\ quad \ text {where} e \ where} \ Mathcal {l}（e，e^\ bot），$$从切线捆绑包的总空间中诱导双分析培养，$ \ mathbf {t}（\ Mathscr {g}（k}（k）（k）（k）$，$ \ mathscr {g}（k）$ \ e $ et $ \ e \ e}（$ e}（$ e}（k）$}（k）（k） $ e $，$ p_e \ in \ mathcal {l}（k）$的正交补充是对$ e $的正交投影，而$ t^*$是$ t $的伴奏）。请注意，此双分析培训是每个切线平面上的仿射图。
Let $X$ be a (real or complex) Banach space, and $\mathcal{I}(X)$ be the set of all (non-zero and non-identity) idempotents; i.e., bounded linear operators on $X$ whose squares equal themselves. We show that the Banach submanifold $\mathcal{I}(X)$ of $\mathcal{L}(X)$ is a locally trivial analytic affine-Banach bundle over the Grassmann manifold $\mathscr{G}(X)$, via the map $κ$ that sends $Q\in \mathcal{I}(X)$ to $Q(X)$, such that the affine-Banach space structure on each fiber is the one induced from $\mathcal{L}(X)$ (in particular, every fiber is an affine-Banach subspace of $\mathcal{L}(X)$). Using this, we show that if $K$ is a real Hilbert space, then the assignment $$(E,T)\mapsto T^*\circ P_{E^\bot} + P_{E}, \quad \text{ where } E\in \mathscr{G}(K)\text{ and } T\in \mathcal{L}(E,E^\bot),$$ induces a bi-analytic bijection from the total space of the tangent bundle, $\mathbf{T}(\mathscr{G}(K))$, of $\mathscr{G}(K)$ onto $\mathcal{I}(K)$ (here, $E^\bot$ is the orthogonal complement of $E$, $P_E\in \mathcal{L}(K)$ is the orthogonal projection onto $E$, and $T^*$ is the adjoint of $T$). Notice that this bi-analytic bijection is an affine map on each tangent plane.
