تقرير
Analytic bundle structure on the idempotent manifold
| العنوان: | Analytic bundle structure on the idempotent manifold |
|---|---|
| المؤلفون: | Leung, Chi-Wai, Ng, Chi-Keung |
| سنة النشر: | 2020 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Differential Geometry, Mathematics - Functional Analysis, Mathematics - Geometric Topology |
| الوصف: | 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 $\kappa$ 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. Comment: Any comment is welcome, especially information on whether the representation results of the tangent bundle of the Grassmannian in Theorem 2 is known in the finite dimensional case |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2001.02352 |
| رقم الأكسشن: | edsarx.2001.02352 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |