Path-connectedness and topological closure of some sets related to the non-compact Stiefel manifold

TL;DR

This paper investigates the topological properties of non-compact Stiefel manifolds in Hilbert spaces, proving results on path-connectedness and closure, with applications to frame theory and solutions of linear equations.

Contribution

It establishes new topological results for non-compact Stiefel manifolds, including path-connectedness and closure properties, with implications for frame theory and continuous Hilbert space frames.

Findings

Proved path-connectedness of certain intersections involving Stiefel manifolds.

Showed the topological closure of Stiefel manifolds contains all polynomial paths passing through points.

Demonstrated the relative density of Stiefel manifolds in specific subsets related to frame theory.

Abstract

If $H$ is a Hilbert space, the non-compact Stiefel manifold $St(n,H)$ consists of independent $n$-tuples in $H$. In this article, we contribute to the topological study of non-compact Stiefel manifolds, mainly by proving two results on the path-connectedness and topological closure of some sets related to the non-compact Stiefel manifold. In the first part, after introducing and proving an essential lemma, we prove that $\bigcap_{j \in J} \left( U(j) + St(n,H) \right)$ is path-connected by polygonal paths under a condition on the codimension of the span of the components of the translating $J$-family. Then, in the second part, we show that the topological closure of $St(n,H) \cap S$ contains all polynomial paths contained in $S$ and passing through a point in $St(n,H)$. As a consequence, we prove that $St(n,H)$ is relatively dense in a certain class of subsets which we illustrate with many examples from frame theory coming from the study of the solutions of some linear and quadratic equations which are finite-dimensional continuous frames. Since $St(n,L^2(X,\mu;\mathbb{F}))$ is isometric to $\mathcal{F}_{(X,\Sigma,\mu),n}^\mathbb{F}$, this article is also a contribution to the theory of finite-dimensional continuous Hilbert space frames.