Topology and Smooth Structure for Pseudoframes

Abstract

Given a closed subspace ${\mathcal{S}}$ of a Hilbert space ${\mathcal{H}}$, we study the sets ${\mathcal{F}_\mathcal{S}}$ of pseudo-frames, ${\mathcal{C}\mathcal{F}_\mathcal{S}}$ of commutative pseudo-frames and ${\tiny{\mathfrak{X}}_{\mathcal{S}}}$ of dual frames for ${\mathcal{S}}$, via the (well known) one to one correspondence which assigns a pair of operators (F, H) to a frame pair ${(\{f_n\}_{n\in\mathbb{N}},\{h_n\}_{n\in\mathbb{N}})}$,

$$F:\ell^2\to\,\mathcal{H}, \quad F\left(\{c_n\}_{n\in\mathbb{N}} \right)=\sum_n c_n f_n,$$

and

$$H:\ell^2 \to\,\mathcal{H}, \quad H=(\{c_n\}_{n\in\mathbb{N}})=\sum_n c_n h_n.$$

We prove that, with this identification, the sets ${\mathcal{F}_\mathcal{S}}$, ${\mathcal{C}\mathcal{F}_\mathcal{S}}$ and ${\tiny{\mathfrak{X}}_{\mathcal{S}}}$ are complemented submanifolds of ${\mathcal{B}(\ell^2,\mathcal{H})\times \mathcal{B}(\ell^2,\mathcal{H})}$. We examine in more detail ${\tiny{\mathfrak{X}}_{\mathcal{S}}}$, which carries a locally transitive action from the general linear group GL(ℓ ²). For instance, we characterize the homotopy theory of ${\tiny{\mathfrak{X}}_{\mathcal{S}}}$ and we prove that ${\tiny{\mathfrak{X}}_{\mathcal{S}}}$ is a strong deformation retract both of ${\mathcal{F}_\mathcal{S}}$ and ${\mathcal{C}\mathcal{F}_\mathcal{S}}$ ; therefore these sets share many of their topological properties.